3 Detailed lookup table and classification for each ODE

The following table 3 gives the classification of each ODE from Maple ODE advisor with a link to each ODE page as well. Clicking on the problem opens a new page that shows the result.

Table 3: ODE classification and performance for each differential equation
#

result

ODE 1

\(y'(x)=a f(x)\)

[_quadrature]

Solution method Separable ODE, Dependent variable missing

Maple
Mathematica

ODE 2

\(y'(x)=y(x)+x+\sin (x)\)

[[_linear, `class A`]]

Solution method Linear ODE

Maple
Mathematica

ODE 3

\(y'(x)=x^2+2 y(x)+3 \cosh (x)\)

[[_linear, `class A`]]

Solution method Linear ODE

Maple
Mathematica

ODE 4

\(y'(x)=a+b x+c y(x)\)

[[_linear, `class A`]]

Solution method Linear ODE

Maple
Mathematica

ODE 5

\(y'(x)=a \cos (b x+c)+k y(x)\)

[[_linear, `class A`]]

Solution method Linear ODE

Maple
Mathematica

ODE 6

\(y'(x)=a \sin (b x+c)+k y(x)\)

[[_linear, `class A`]]

Solution method Linear ODE

Maple
Mathematica

ODE 7

\(y'(x)=a+b e^{k x}+c y(x)\)

[[_linear, `class A`]]

Solution method Linear ODE

Maple
Mathematica

ODE 8

\(y'(x)=x \left (x^2-y(x)\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 9

\(y'(x)=x \left (a y(x)+e^{-x^2}\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 10

\(y'(x)=x^2 \left (a x^3+b y(x)\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 11

\(y'(x)=a x^n y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 12

\(y'(x)=y(x) \cos (x)+\sin (x) \cos (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 13

\(y'(x)=y(x) \cos (x)+e^{\sin (x)}\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 14

\(y'(x)=y(x) \cot (x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 15

\(y'(x)=1-y(x) \cot (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 16

\(y'(x)=x \csc (x)-y(x) \cot (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 17

\(y'(x)=y(x) (\cot (x)+2 \csc (2 x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 18

\(y'(x)=\sec (x)-y(x) \cot (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 19

\(y'(x)=y(x) \cot (x)+e^x \sin (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 20

\(y'(x)+2 y(x) \cot (x)+\csc (x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 21

\(y'(x)=4 x \csc (x) \sec ^2(x)-2 y(x) \cot (2 x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 22

\(y'(x)=2 \left (\cos (2 x) \cot ^2(x)-y(x) \csc (2 x)\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 23

\(y'(x)=4 x \csc (x) \left (y(x)+\sin ^3(x)\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 24

\(y'(x)=4 x \csc (x) \left (y(x)-\tan ^2(x)+1\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 25

\(y'(x)=y(x) \sec (x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 26

\(y'(x)+\tan (x)=(1-y(x)) \sec (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 27

\(y'(x)=y(x) \tan (x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 28

\(y'(x)=y(x) \tan (x)+\cos (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 29

\(y'(x)=\cos (x)-y(x) \tan (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 30

\(y'(x)=\sec (x)-y(x) \tan (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 31

\(y'(x)=y(x) \tan (x)+\sin (2 x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 32

\(y'(x)=\sin (2 x)-y(x) \tan (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 33

\(y'(x)=2 y(x) \tan (x)+\sin (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 34

\(y'(x)=2 (y(x) \tan (2 x)+\sec (2 x)+1)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 35

\(y'(x)=3 y(x) \tan (x)+\csc (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 36

\(y'(x)=y(x) (a+\sin (\log (x))+\cos (\log (x)))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 37

\(y'(x)=6 e^{2 x}-y(x) \tanh (x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 38

\(y'(x)=y(x) f'(x)+f(x) f'(x)\)

odeadvisor timed out

Solution method Linear ODE

Maple
Mathematica

ODE 39

\(y'(x)=f(x)+g(x) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 40

\(y'(x)=x^2-y(x)^2\)

[_Riccati]

Solution method Series solution to \(y'(x)=f(x,y(x))\), case \(f(x,y)\) analytic

Maple
Mathematica

ODE 41

\(f(x)^2+y'(x)=f'(x)+y(x)^2\)

odeadvisor timed out

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 42

\(y'(x)-x+1=y(x) (y(x)+x)\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 43

\(y'(x)=(y(x)+x)^2\)

[[_homogeneous, `class C`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 44

\(y'(x)=(x-y(x))^2\)

[[_homogeneous, `class C`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 45

\(y'(x)=(x-y(x))^2+3 (y(x)-x+1)\)

[[_homogeneous, `class C`], _Riccati]

Solution method Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Maple
Mathematica

ODE 46

\(y'(x)=-\left (x^2+1\right ) y(x)+y(x)^2+2 x\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 47

\(y'(x)=x \left (x^3+2\right )-\left (2 x^2-y(x)\right ) y(x)\)

[[_1st_order, _with_linear_symmetries], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 48

\(y'(x)=x \left (2-x^3\right )+\left (2 x^2-y(x)\right ) y(x)+1\)

[[_1st_order, _with_linear_symmetries], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 49

\(y'(x)=\cos (x)-y(x) (\sin (x)-y(x))\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 50

\(y'(x)=y(x) (y(x)+\sin (2 x))+\cos (2 x)\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 51

\(y'(x)=x f(x) y(x)+f(x)+y(x)^2\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 52

\(y'(x)=(-4 y(x)+x+3)^2\)

[[_homogeneous, `class C`], _Riccati]

Solution method Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Maple
Mathematica

ODE 53

\(y'(x)=(9 y(x)+4 x+1)^2\)

[[_homogeneous, `class C`], _Riccati]

Solution method Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Maple
Mathematica

ODE 54

\(y'(x)=3 \left (a+b y(x)^2+b x\right )\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 55

\(y'(x)=a+b y(x)^2\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 56

\(y'(x)=a x+b y(x)^2\)

[[_Riccati, _special]]

Solution method Riccati ODE, Main form

Maple
Mathematica

ODE 57

\(y'(x)=a+b x+c y(x)^2\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 58

\(y'(x)=a x^{n-1}+b x^{2 n}+c y(x)^2\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 59

\(y'(x)=a x^2+b y(x)^2\)

[[_Riccati, _special]]

Solution method Riccati ODE, Main form

Maple
Mathematica

ODE 60

\(y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 61

\(y'(x)=a y(x)+b y(x)^2+f(x)\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 62

\(y'(x)=a (x-y(x)) y(x)+1\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 63

\(y'(x)=a y(x)^2+f(x)+g(x) y(x)\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 64

\(y'(x)=x y(x) (y(x)+3)\)

[_separable]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 65

\(y'(x)=-x^3+\left (2 x^2+1\right ) y(x)-x y(x)^2-x+1\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 66

\(y'(x)=x \left (x^2 y(x)-y(x)^2+2\right )\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 67

\(y'(x)=-\left ((1-x) y(x)^2\right )+(1-2 x) y(x)+x\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 68

\(y'(x)=a x y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 69

\(y'(x)=x^n \left (a+b y(x)^2\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 70

\(y'(x)=a x^m+b x^n y(x)^2\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 71

\(y'(x)=y(x) (a+b y(x) \cos (k x))\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 72

\(y'(x)=\sin (x) \left (2 \sec ^2(x)-y(x)\right )\)

[_linear]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 73

\(y'(x)+4 \csc (x)=y(x)^2 \sin (x)+y(x) (3-\cot (x))\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 74

\(y'(x)=y(x) \sec (x)+(\sin (x)-1)^2\)

[_linear]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 75

\(y'(x)+\left (1-y(x)^2\right ) \tan (x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 76

\(y'(x)=f(x)+g(x) y(x)+h(x) y(x)^2\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 77

\(y'(x)=f(x) \left (a+b y(x)+c y(x)^2\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 78

\(y(x)^2 (a x+y(x))+y'(x)\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 79

\(y'(x)=y(x)^2 \left (a e^x+y(x)\right )\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 80

\(3 a (y(x)+2 x) y(x)^2+y'(x)=0\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 81

\(y'(x)=y(x) \left (a+b y(x)^2\right )\)

[_quadrature]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 82

\(y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 83

\(y'(x)=x y(x)^3\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 84

\(y'(x)+y(x) \left (1-x y(x)^2\right )=0\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 85

\(y'(x)=y(x)^2 (a+b x y(x))\)

[[_homogeneous, `class G`], _Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 86

\(y(x)^3 \left (a+4 b^2 x+3 b x^2\right )+y'(x)+3 x y(x)^2=0\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 87

\(y'(x)=y(x)^2 \left (x^3 y(x)+1\right )\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 88

\(2 x y(x) \left (a x y(x)^2+1\right )+y'(x)=0\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 89

\(y'(x)=y(x)^2-a x \left (1-x^{n-1}\right ) y(x)^3\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 90

\(y'(x)=a y(x)^2+x y(x)^3 \left (b+c x^{n-1}\right )\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 91

\(y'(x)+y(x) \left (y(x)^2 \sec (x)+\tan (x)\right )=0\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 92

\(y'(x)+y(x)^3 \tan (x) \sec (x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 93

\(y'(x)=\text {f0}(x)+\text {f1}(x) y(x)+\text {f2}(x) y(x)^2+\text {f3}(x) y(x)^3\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 94

\(y'(x)=y(x) \left (y(x)^3 \sec (x)+\tan (x)\right )\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 95

\(y'(x)=a x^{\frac {n}{1-n}}+b y(x)^n\)

[[_homogeneous, `class G`], _Chini]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 96

\(y'(x)=f(x) y(x)+g(x) y(x)^k\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 97

\(y'(x)=f(x)+g(x) y(x)+h(x) y(x)^n\)

[_Chini]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 98

\(y'(x)=f(x) y(x)^m+g(x) y(x)^n\)

[NONE]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 99

\(y'(x)=\sqrt {| y(x)| }\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 100

\(y'(x)=a+\sqrt {\text {A0}+\text {B0} y(x)}+b y(x)\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 101

\(y'(x)=a x+b \sqrt {y(x)}\)

[[_homogeneous, `class G`], _Chini]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 102

\(x^3+y'(x)=x \sqrt {x^4+4 y(x)}\)

[[_1st_order, _with_linear_symmetries]]

Solution method Homogeneous equation, isobaric equation

Maple
Mathematica

ODE 103

\(y'(x)+2 \left (1-x \sqrt {y(x)}\right ) y(x)=0\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 104

\(y'(x)=\sqrt {a+b y(x)^2}\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 105

\(y'(x)=y(x) \sqrt {a+b y(x)}\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 106

\(g(x) (f(x)-y(x)) \sqrt {(y(x)-a) (y(x)-b)}+y'(x)=0\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 107

\(y'(x)=\sqrt {X Y}\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 108

\(y'(x)=\text {R1}\left (x,\sqrt {X}\right ) \text {R2}\left (y(x),\sqrt {Y}\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 109

\(y'(x)=\cos ^2(x) \cos (y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 110

\(y'(x)=\sec ^2(x) \text {Cosy}(y(x)) \cot (y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 111

\(y'(x)=a+b \cos (A x+B y(x))\)

[[_homogeneous, `class C`], _dAlembert]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 112

\(y'(x)=-\left (1-f'(x)\right ) \cos (y(x))+f'(x)-f(x) \sin (y(x))+1\)

odeadvisor timed out

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 113

\(g(x) \sin (a y(x))+h(x) \cos (a y(x))+f(x)+y'(x)=0\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 114

\(y'(x)=a+b \cos (y(x))\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 115

\(x \left (\sin (2 y(x))-x^2 \cos ^2(y(x))\right )+y'(x)=0\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 116

\(y'(x)+\tan (x) \sec (x) \cos ^2(y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 117

\(y'(x)=\cot (x) \cot (y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 118

\(y'(x)+\cot (x) \cot (y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 119

\(y'(x)=\sin (x) (\csc (y(x))-\cot (y(x)))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 120

\(y'(x)=\tan (x) \cot (y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 121

\(y'(x)+\tan (x) \cot (y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 122

\(y'(x)+\sin (2 x) \csc (2 y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 123

\(y'(x)=\tan (x) (\tan (y(x))+\sec (x) \sec (y(x)))\)

[`y=_G(x,y')`]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 124

\(y'(x)=\cos (x) \sec ^2(y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 125

\(y'(x)=\sec ^2(x) \sec ^3(y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 126

\(y'(x)=a+b \sin (y(x))\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 127

\(y'(x)=a+b \sin (A x+B y(x))\)

[[_homogeneous, `class C`], _dAlembert]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 128

\(y'(x)=\tan (y(x)) (\cos (x) \sin (y(x))+1)\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 129

\(y'(x)+\csc (2 x) \sin (2 y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 130

\(f(x)+g(x) \tan (y(x))+y'(x)=0\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 131

\(y'(x)=\sqrt {a+b \cos (y(x))}\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 132

\(y'(x)=e^{y(x)}+x\)

[[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

Solution method Series solution to \(y'(x)=f(x,y(x))\), case \(f(x,y)\) analytic

Maple
Mathematica

ODE 133

\(y'(x)=e^{y(x)+x}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 134

\(y'(x)=e^x \left (a+b e^{-y(x)}\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 135

\(y'(x)+y(x) \log (x) \log (y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 136

\(y'(x)=x^{m-1} y(x)^{1-n} f\left (a x^m+b y(x)^n\right )\)

[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 137

\(y'(x)=a f(y(x))\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 138

\(y'(x)=f(a+b x+c y(x))\)

[[_homogeneous, `class C`], _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Maple
Mathematica

ODE 139

\(y'(x)=f(x) g(y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 140

\(y'(x)=\text {Csx}(x) y(x) \sec (x)+\sec ^2(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 141

\(2 y'(x)+2 \csc ^2(x)=y(x) \csc (x) \sec (x)-y(x)^2 \sec ^2(x)\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 142

\(2 y'(x)=2 \sin ^2(y(x)) \tan (y(x))-x \sin (2 y(x))\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 143

\(a x+2 y'(x)=\sqrt {a^2 x^2-4 b x^2-4 c y(x)}\)

[[_homogeneous, `class G`]]

Solution method Homogeneous equation, isobaric equation

Maple
Mathematica

ODE 144

\(3 y'(x)=\sqrt {x^2-3 y(x)}+x\)

[[_1st_order, _with_linear_symmetries], _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 145

\(x y'(x)=\sqrt {a^2-x^2}\)

[_quadrature]

Solution method Separable ODE, Dependent variable missing

Maple
Mathematica

ODE 146

\(x y'(x)+y(x)+x=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 147

\(x^2+x y'(x)-y(x)=0\)

[_linear]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 148

\(x y'(x)=x^3-y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 149

\(x y'(x)=x^3+y(x)+1\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 150

\(x y'(x)=x^m+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 151

\(x y'(x)=x \sin (x)-y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 152

\(x y'(x)=x^2 \sin (x)+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 153

\(x y'(x)=x^n \log (x)-y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 154

\(x y'(x)=\sin (x)-2 y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 155

\(x y'(x)=a y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 156

\(x y'(x)=a y(x)+x+1\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 157

\(x y'(x)=a x+b y(x)\)

[_linear]

Solution method Homogeneous equation

Maple
Mathematica

ODE 158

\(x y'(x)=a x^2+b y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 159

\(x y'(x)=a+b x^n+c y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 160

\(x y'(x)+(3-x) y(x)+2=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 161

\((a x+2) y(x)+x y'(x)+x=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 162

\(y(x) (a+b x)+x y'(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 163

\(x y'(x)=x^3+\left (1-2 x^2\right ) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 164

\(x y'(x)=a x-\left (1-b x^2\right ) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 165

\(\left (2-a x^2\right ) y(x)+x y'(x)+x=0\)

[_linear]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 166

\(x^2+x y'(x)+y(x)^2=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 167

\(x y'(x)=x^2+y(x) (y(x)+1)\)

[[_homogeneous, `class D`], _rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 168

\(x y'(x)+y(x)^2-y(x)=x^{2/3}\)

[_rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 169

\(x y'(x)=a+b y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 170

\(x y'(x)=a x^2+b y(x)^2+y(x)\)

[[_homogeneous, `class D`], _rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 171

\(x y'(x)=a x^{2 n}+y(x) (b y(x)+n)\)

[_rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 172

\(x y'(x)=a x^n+b y(x)+c y(x)^2\)

[_rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 173

\(x y'(x)=a x^n+b y(x)+c y(x)^2+k\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 174

\(a+x y'(x)+x y(x)^2=0\)

[_rational, [_Riccati, _special]]

Solution method Riccati ODE, Main form

Maple
Mathematica

ODE 175

\(x y'(x)+y(x) (1-x y(x))=0\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 176

\(x y'(x)=y(x) (1-x y(x))\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 177

\(x y'(x)=y(x) (x y(x)+1)\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 178

\(x y'(x)=a x^3 y(x) (1-x y(x))\)

[_Bernoulli]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 179

\(x y'(x)=x^3+\left (2 x^2+1\right ) y(x)+x y(x)^2\)

[[_homogeneous, `class D`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 180

\(x y'(x)=y(x) (2 x y(x)+1)\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 181

\(y(x) (a x y(x)+2)+b x+x y'(x)=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 182

\(\text {a0}+\text {a1} x+y(x) (\text {a2}+\text {a3} x y(x))+x y'(x)=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 183

\(a x^2 y(x)^2+x y'(x)+2 y(x)=b\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 184

\(\frac {1}{2} (n-m) y(x)+x^m+x^n y(x)^2+x y'(x)=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 185

\(y(x) \left (a+b x^n y(x)\right )+x y'(x)=0\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 186

\(x y'(x)=a x^m-b y(x)-c x^n y(x)^2\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 187

\(x y'(x)=a x^n (x-y(x))^2-y(x)+2 x\)

[[_1st_order, _with_linear_symmetries], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 188

\(y(x) (1-a y(x) \log (x))+x y'(x)=0\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 189

\(x y'(x)=f(x) \left (x^2-y(x)^2\right )+y(x)\)

[[_homogeneous, `class D`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 190

\(x y'(x)=y(x) \left (y(x)^2+1\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 191

\(x y'(x)+y(x) \left (1-x y(x)^2\right )=0\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 192

\(x y'(x)+y(x)=a \left (x^2+1\right ) y(x)^3\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 193

\(x y'(x)=a y(x)+b \left (x^2+1\right ) y(x)^3\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 194

\(x y'(x)+2 y(x)=a x^{2 k} y(x)^k\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 195

\(x y'(x)=4 \left (y(x)-\sqrt {y(x)}\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 196

\(x y'(x)+2 y(x)=\sqrt {y(x)^2+1}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 197

\(x y'(x)=\sqrt {x^2+y(x)^2}+y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 198

\(x y'(x)=\sqrt {x^2-y(x)^2}+y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 199

\(x y'(x)=x \sqrt {x^2+y(x)^2}+y(x)\)

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Homogeneous equation, \(xy'(x)=x f(x) g(u)+y(x)\)

Maple
Mathematica

ODE 200

\(x y'(x)=y(x)-x (x-y(x)) \sqrt {x^2+y(x)^2}\)

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Homogeneous equation, \(xy'(x)=x f(x) g(u)+y(x)\)

Maple
Mathematica

ODE 201

\(x y'(x)=a \sqrt {b^2 x^2+y(x)^2}+y(x)\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 202

\(\cos (y(x)) \left (\sin (y(x))-3 x^2 \cos (y(x))\right )+x y'(x)=0\)

[`y=_G(x,y')`]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 203

\(x y'(x)-y(x)+x \cos \left (\frac {y(x)}{x}\right )+x=0\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 204

\(x y'(x)=y(x)-x \cos ^2\left (\frac {y(x)}{x}\right )\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 205

\(x y'(x)=\left (1-2 x^2\right ) \cot ^2(y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 206

\(x y'(x)=y(x)-\cot ^2(y(x))\)

[_separable]

Solution method Homogeneous equation

Maple
Mathematica

ODE 207

\(x y'(x)+y(x)+2 x \sec (x y(x))=0\)

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 208

\(x y'(x)-y(x)+x \sec \left (\frac {y(x)}{x}\right )=0\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 209

\(x y'(x)=y(x)+x \sec ^2\left (\frac {y(x)}{x}\right )\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 210

\(x y'(x)=\sin (x-y(x))\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 211

\(x y'(x)=y(x)+x \sin \left (\frac {y(x)}{x}\right )\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 212

\(x y'(x)+\tan (y(x))=0\)

[_separable]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 213

\(x y'(x)+\tan (y(x)+x)+x=0\)

[[_1st_order, _with_linear_symmetries]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 214

\(x y'(x)=y(x)-x \tan \left (\frac {y(x)}{x}\right )\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 215

\(x y'(x)=\left (y(x)^2+1\right ) \left (x^2+\tan ^{-1}(y(x))\right )\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 216

\(x y'(x)=x e^{\frac {y(x)}{x}}+y(x)\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 217

\(x y'(x)=x e^{\frac {y(x)}{x}}+y(x)+x\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 218

\(x y'(x)=y(x) \log (y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 219

\(x y'(x)=y(x) (-\log (y(x))+\log (x)+1)\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 220

\(x y'(x)+y(x) (-\log (y(x))-\log (x)+1)=0\)

[[_homogeneous, `class G`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 221

\(x y'(x)=y(x)-2 x \tanh \left (\frac {y(x)}{x}\right )\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 222

\(n y(x)+x y'(x)=f(x) g\left (x^n y(x)\right )\)

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 223

\(x y'(x)=y(x) f\left (x^m y(x)^n\right )\)

[[_homogeneous, `class G`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 224

\((x+1) y'(x)=(3 x+4) x^3+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 225

\((x+1) y'(x)=2 y(x)+(x+1)^4\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 226

\((x+1) y'(x)=n y(x)+e^x (x+1)^{n+1}\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 227

\((x+1) y'(x)=a y(x)+b x y(x)^2\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 228

\((x+1) y'(x)+(x+1)^4 y(x)^3+y(x)=0\)

[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 229

\((x+1) y'(x)=y(x) \left (1-x y(x)^3\right )\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 230

\((x+1) y'(x)=(x+1) \sqrt {y(x)+1}+y(x)+1\)

[[_1st_order, _with_linear_symmetries]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 231

\((a+x) y'(x)=b x\)

[_quadrature]

Solution method Separable ODE, Dependent variable missing

Maple
Mathematica

ODE 232

\((a+x) y'(x)=b x+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 233

\((a+x) y'(x)+b x^2+y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 234

\((a+x) y'(x)=2 (a+x)^5+3 y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 235

\((a+x) y'(x)=b+c y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 236

\((a+x) y'(x)=b x+c y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 237

\((a+x) y'(x)=y(x) (1-a y(x))\)

[_separable]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 238

\((a-x) y'(x)=y(x)^3 (b+c x)+y(x)\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 239

\(2 x y'(x)=2 x^3-y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 240

\(2 x y'(x)+1=y(x)^2+4 i x y(x)\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 241

\(2 x y'(x)=y(x) \left (y(x)^2+1\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 242

\(2 x y'(x)+y(x) \left (y(x)^2+1\right )=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 243

\(2 x y'(x)=y(x) \left (-6 y(x)^2+x+1\right )\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 244

\(\sqrt {a^2-4 b-4 c y(x)}+a+2 x y'(x)+4 y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 245

\((1-2 x) y'(x)=2 (-3 y(x)+16 x+8)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 246

\((2 x+1) y'(x)=4 e^{-y(x)}-2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 247

\(2 (1-x) y'(x)=y(x)+4 \sqrt {1-x} x\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 248

\(2 (x+1) y'(x)+(x+1)^4 y(x)^3+2 y(x)=0\)

[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 249

\(3 x y'(x)=3 x^{2/3}+(1-3 y(x)) y(x)\)

[_rational, _Riccati]

Solution method Riccati ODE, Special cases

Maple
Mathematica

ODE 250

\(3 x y'(x)=y(x) \left (x y(x)^3+2\right )\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 251

\(3 x y'(x)=y(x) \left (3 x y(x)^3 \log (x)+1\right )\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 252

\(x^2 y'(x)=a-y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 253

\(x^2 y'(x)=a+b x+c x^2+x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 254

\(x^2 y'(x)=a+b x+c x^2-x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 255

\(x^2 y'(x)+(1-2 x) y(x)=x^2\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 256

\(x^2 y'(x)=a+b x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 257

\(x^2 y'(x)=y(x) (a+b x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 258

\(x^2 y'(x)+(x+2) x y(x)=\left (1-e^{-2 x}\right ) x-2\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 259

\(x^2 y'(x)+2 (1-x) x y(x)=e^x \left (2 e^x-1\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 260

\(x^2 y'(x)+x^2+x y(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _rational, _Riccati]

Solution method Homogeneous equation

Maple
Mathematica

ODE 261

\(x^2 y'(x)=(-y(x)+2 x+1)^2\)

[[_homogeneous, `class C`], _rational, _Riccati]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 262

\(x^2 y'(x)=a+b y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 263

\(x^2 y'(x)=y(x) (a y(x)+x)\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 264

\(x^2 y'(x)=y(x) (a x+b y(x))\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 265

\(a x^2+b x y(x)+c y(x)^2+x^2 y'(x)=0\)

[[_homogeneous, `class A`], _rational, _Riccati]

Solution method Homogeneous equation

Maple
Mathematica

ODE 266

\(x^2 y'(x)=a+b x^n+x^2 y(x)^2\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 267

\(x^2 y'(x)+x y(x) (x y(x)+4)+2=0\)

[[_homogeneous, `class G`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 268

\(a x (1-x y(x))+x^2 y'(x)+x^2 \left (-y(x)^2\right )+2=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 269

\(x^2 y'(x)=a+b x^2 y(x)^2\)

[[_homogeneous, `class G`], _rational, [_Riccati, _special]]

Solution method Riccati ODE, Main form

Maple
Mathematica

ODE 270

\(x^2 y'(x)=a+b x^n+c x^2 y(x)^2\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 271

\(x^2 y'(x)=a+b x y(x)+c x^2 y(x)^2\)

[[_homogeneous, `class G`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 272

\(x^2 y'(x)=a+b x y(x)+c x^4 y(x)^2\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 273

\(x^2 y'(x)+y(x) \left (x^2+y(x)^2-x\right )=0\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 274

\(x^2 y'(x)=2 y(x) \left (x-y(x)^2\right )\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 275

\(x^2 y'(x)=a x^2 y(x)^2-a y(x)^3\)

[_rational, _Abel]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 276

\(a y(x)^2+b x^2 y(x)^3+x^2 y'(x)=0\)

[_rational, _Abel]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 277

\(x^2 y'(x)=y(x) \left (a x+b y(x)^3\right )\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 278

\(x^2 y'(x)+x y(x)+\sqrt {y(x)}=0\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 279

\(x^2 y'(x)=3 x \tan (y(x))+\sec (y(x))\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 280

\(\left (1-x^2\right ) y'(x)=-x^2+y(x)+1\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 281

\(\left (1-x^2\right ) y'(x)+1=x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 282

\(\left (1-x^2\right ) y'(x)=5-x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 283

\(a+\left (x^2+1\right ) y'(x)+x y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 284

\(a+\left (x^2+1\right ) y'(x)-x y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 285

\(a+\left (1-x^2\right ) y'(x)-x y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 286

\(\left (1-x^2\right ) y'(x)+x y(x)-x=0\)

[_separable]

Solution method Linear ODE

Maple
Mathematica

ODE 287

\(\left (1-x^2\right ) y'(x)-x^2+x y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 288

\(\left (1-x^2\right ) y'(x)+x^2+x y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 289

\(\left (x^2+1\right ) y'(x)=x \left (x^2+1\right )-x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 290

\(\left (x^2+1\right ) y'(x)=x \left (3 x^2-y(x)\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 291

\(\left (1-x^2\right ) y'(x)+2 x y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 292

\(\left (x^2+1\right ) y'(x)=2 x (x-y(x))\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 293

\(\left (x^2+1\right ) y'(x)=2 x \left (x^2+1\right )^2+2 x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 294

\(\left (1-x^2\right ) y'(x)+\cos (x)=2 x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 295

\(\left (x^2+1\right ) y'(x)=\tan (x)-2 x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 296

\(\left (1-x^2\right ) y'(x)=a+4 x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 297

\(\left (x^2+1\right ) y'(x)=y(x) (a+2 b x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 298

\(\left (x^2+1\right ) y'(x)=y(x)^2+1\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 299

\(\left (1-x^2\right ) y'(x)=1-y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 300

\(\left (1-x^2\right ) y'(x)=1-(2 x-y(x)) y(x)\)

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 301

\(\left (1-x^2\right ) y'(x)=n \left (y(x)^2-2 x y(x)+1\right )\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 302

\(\left (x^2+1\right ) y'(x)+x (1-y(x)) y(x)=0\)

[_separable]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 303

\(\left (1-x^2\right ) y'(x)=x y(x) (a y(x)+1)\)

[_separable]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 304

\(\left (x^2+1\right ) y'(x)=y(x)^2-2 x \left (y(x)^2+1\right ) y(x)+1\)

[_rational, _Abel]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 305

\(\left (x^2+1\right ) y'(x)+x \sin (y(x)) \cos (y(x))=x \left (x^2+1\right ) \cos ^2(y(x))\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 306

\(\left (x^2+1\right ) y'(x)=x^2-y(x) \cot ^{-1}(x)+1\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 307

\(\left (4-x^2\right ) y'(x)+4 y(x)=(x+2) y(x)^2\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 308

\(\left (a^2+x^2\right ) y'(x)=b+x y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 309

\(\left (a^2+x^2\right ) y'(x)=\left (\sqrt {a^2+x^2}+x\right ) (b+y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 310

\(\left (a^2+x^2\right ) y'(x)+(x-y(x)) y(x)=0\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 311

\(\left (a^2+x^2\right ) y'(x)=a^2-2 y(x)^2+3 x y(x)\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 312

\(\left (a^2+x^2\right ) y'(x)+b x y(x)^2+x y(x)=0\)

[_separable]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 313

\((1-x) x y'(x)=a+(x+1) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 314

\((1-x) x y'(x)=2 (x y(x)+1)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 315

\((1-x) x y'(x)=2 (x y(x)-1)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 316

\(x (x+1) y'(x)=(1-2 x) y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 317

\((1-x) x y'(x)+(2 x+1) y(x)=a\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 318

\((1-x) x y'(x)=a+2 (2-x) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 319

\((1-x) x y'(x)-3 x y(x)+y(x)+2=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 320

\(x (x+1) y'(x)=\left (x^2+x-1\right ) y(x)+(x+1) \left (x^2-1\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 321

\(x^2+(x-3) (x-2) y'(x)+3 x y(x)-8 y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 322

\(x (a+x) y'(x)=y(x) (b+c y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 323

\((a+x)^2 y'(x)=2 (a+x) (b+y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 324

\(k (-a+y(x)+x)^2+(x-a)^2 y'(x)+y(x)^2=0\)

[[_homogeneous, `class C`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 325

\((x-a) (x-b) y'(x)+k y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 326

\((x-a) (x-b) y'(x)=y(x) (-a-b+2 x)+(x-a) (x-b)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 327

\((x-a) (x-b) y'(x)=c y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 328

\(k (y(x)-a) (y(x)-b)+(x-a) (x-b) y'(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 329

\(k (-a+y(x)+x) (-b+y(x)+x)+(x-a) (x-b) y'(x)+y(x)^2=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 330

\(2 x^2 y'(x)=y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 331

\(2 x^2 y'(x)+2 x^2 y(x) \cot (x)+x \cot (x)-1=0\)

[_linear]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 332

\(2 x^2 y'(x)+x^2 \left (-y(x)^2\right )+2 x y(x)+1=0\)

[[_homogeneous, `class G`], _rational, _Riccati]

Solution method Homogeneous equation, isobaric equation

Maple
Mathematica

ODE 333

\(2 x^2 y'(x)=\left (x^2-y(x)^2\right ) (1-x \cot (x))+2 x y(x)\)

[[_homogeneous, `class D`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 334

\(2 \left (1-x^2\right ) y'(x)=\sqrt {1-x^2}+(x+1) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 335

\((1-2 x) x y'(x)+(1-4 x) y(x)+1=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 336

\((1-2 x) x y'(x)=y(x)^2-(4 x+1) y(x)+4 x\)

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 337

\(2 (1-x) x y'(x)+(1-2 x) y(x)+x=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 338

\(2 (1-x) x y'(x)+(1-x) y(x)^2+x=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 339

\(2 \left (x^2+x+1\right ) y'(x)=8 x^2-(2 x+1) y(x)+1\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 340

\(4 \left (x^2+1\right ) y'(x)-x^2-4 x y(x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 341

\(a x^2 y'(x)=a x y(x)+b^2 y(x)^2+x^2\)

[[_homogeneous, `class A`], _rational, _Riccati]

Solution method Homogeneous equation

Maple
Mathematica

ODE 342

\(\left (a+b x^2\right ) y'(x)=A+B y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 343

\(\left (a+b x^2\right ) y'(x)=c x y(x) \log (y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 344

\(x (a x+1) y'(x)+a-y(x)=0\)

[_separable]

Solution method Linear ODE

Maple
Mathematica

ODE 345

\((a+b x)^2 y'(x)+y(x)^3 (a+b x)+c y(x)^2=0\)

[_rational, _Abel]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 346

\(x^3 y'(x)=a+b x^2 y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 347

\(x^3 y'(x)=x^2 y(x)-x^2+3\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 348

\(x^3 y'(x)=x^4+y(x)^2\)

[[_homogeneous, `class G`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 349

\(x^3 y'(x)=y(x) \left (x^2+y(x)\right )\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 350

\(x^3 y'(x)=x^2 (y(x)-1)+y(x)^2\)

[[_homogeneous, `class D`], _rational, _Riccati]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 351

\(x^3 y'(x)=(x+1) y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 352

\(x^3 y'(x)+x^2 y(x) \left (1-x^2 y(x)\right )+20=0\)

[[_homogeneous, `class G`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 353

\(x^6 \left (-y(x)^2\right )+x^3 y'(x)+(3-2 x) x^2 y(x)+3=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 354

\(x^3 y'(x)=y(x) \left (2 x^2+y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method Homogeneous equation

Maple
Mathematica

ODE 355

\(x^3 y'(x)=\cos (y(x)) \left (\cos (y(x))-2 x^2 \sin (y(x))\right )\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 356

\(x \left (x^2+1\right ) y'(x)=a x^2+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 357

\(x \left (1-x^2\right ) y'(x)=a x^2+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 358

\(x \left (x^2+1\right ) y'(x)=a x^3+y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 359

\(x \left (x^2+1\right ) y'(x)=a-x^2 y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 360

\(x \left (x^2+1\right ) y'(x)=\left (1-x^2\right ) y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 361

\(x \left (1-x^2\right ) y'(x)=\left (x^2-x+1\right ) y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 362

\(x \left (1-x^2\right ) y'(x)=a x^3+\left (1-2 x^2\right ) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 363

\(x \left (1-x^2\right ) y'(x)=\left (1-2 x^2\right ) y(x)+\left (1-x^2\right ) x^3\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 364

\(x \left (x^2+1\right ) y'(x)=2 \left (1-2 x^2 y(x)\right )\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 365

\(x \left (x^2+1\right ) y'(x)=x-\left (5 x^2+3\right ) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 366

\(\left (1-x^2\right ) x y'(x)+\left (1-x^2\right ) y(x)^2+x^2=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 367

\((1-x) x^2 y'(x)=(2-x) x y(x)-y(x)^2\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 368

\(2 x^3 y'(x)=y(x) \left (x^2-y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 369

\(2 x^3 y'(x)=y(x) \left (a y(x)^2+3 x^2\right )\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 370

\(6 x^3 y'(x)=4 x^2 y(x)+(1-3 x) y(x)^4\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 371

\(x y'(x) \left (a+b x+c x^2\right )-y(x) \left (a+b x+c x^2\right )+x^2=y(x)^2\)

[[_homogeneous, `class D`], _rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 372

\(x^4 y'(x)=y(x) \left (x^3+y(x)\right )\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 373

\(a^2+x^4 y'(x)+x^4 y(x)^2=0\)

[_rational, [_Riccati, _special]]

Solution method Riccati ODE, Main form

Maple
Mathematica

ODE 374

\(x^4 y'(x)+x^3 y(x)+\csc (x y(x))=0\)

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 375

\(\left (1-x^4\right ) y'(x)=2 x \left (1-y(x)^2\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 376

\(x \left (1-x^3\right ) y'(x)=2 x-\left (1-4 x^3\right ) y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 377

\(x \left (1-x^3\right ) y'(x)=x^2+y(x) (1-2 x y(x))\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 378

\(x^2 \left (1-x^2\right ) y'(x)=y(x) \left (x-3 x^3 y(x)\right )\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 379

\(x \left (1-2 x^3\right ) y'(x)=2 \left (1-x^3\right ) y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 380

\(\left (y'(x)+y(x)^2\right ) \left (a+b x+c x^2\right )^2+A=0\)

[_rational, _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 381

\(x^5 y'(x)=1-3 x^4 y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 382

\(x \left (1-x^4\right ) y'(x)=\left (1-x^4\right ) y(x)+2 x \left (x^2-y(x)^2\right )\)

[[_homogeneous, `class D`], _rational, _Riccati]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 383

\(x^7 y'(x)+5 x^3 y(x)^2+2 \left (x^2+1\right ) y(x)^3=0\)

[_rational, _Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 384

\(x^n y'(x)=a+b x^{n-1} y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 385

\(x^n y'(x)=x^{2 n-1}-y(x)^2\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 386

\(y(x) \left (x^{n-1}+y(x)\right )+x^n y'(x)+x^2=0\)

[_Riccati]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 387

\((1-n) x^{n-1}+x^{2 n-2}+x^n y'(x)+y(x)^2=0\)

[_Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 388

\(x^n y'(x)=a^2 x^{2 n-2}+b^2 y(x)^2\)

[[_homogeneous, `class G`], _Riccati]

Solution method Riccati ODE, Generalized ODE

Maple
Mathematica

ODE 389

\(x^n y'(x)=x^{n-1} \left (a x^{2 n}-b y(x)^2+n y(x)\right )\)

[_rational, _Riccati]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 390

\(x^{2 n} y'(x)=-n x^{n-1}+x^n y(x) \left (x^{2 n} y(x)^2-3 x^n y(x)+1\right )+1\)

[_Abel]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 391

\(x^k y'(x)=a x^m+b y(x)^n\)

[_Chini]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 392

\(\sqrt {x^2+1} y'(x)=2 x-y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 393

\(\sqrt {1-x^2} y'(x)=y(x)^2+1\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 394

\(\left (x-\sqrt {x^2+1}\right ) y'(x)=y(x)+\sqrt {y(x)^2+1}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 395

\(\sqrt {a^2+x^2} y'(x)+y(x)+x=\sqrt {a^2+x^2}\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 396

\(\sqrt {b^2+x^2} y'(x)=\sqrt {a^2+y(x)^2}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 397

\(\sqrt {b^2-x^2} y'(x)=\sqrt {a^2-y(x)^2}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 398

\(x \sqrt {a^2+x^2} y'(x)=y(x) \sqrt {b^2+y(x)^2}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 399

\(x \sqrt {x^2-a^2} y'(x)=y(x) \sqrt {y(x)^2-b^2}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 400

\(\sqrt {X} y'(x)+\sqrt {Y}=0\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 401

\(\sqrt {X} y'(x)=\sqrt {Y}\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 402

\(x^{3/2} y'(x)=a+b x^{3/2} y(x)^2\)

[_rational, [_Riccati, _special]]

Solution method Abel ODE, First kind

Maple
Mathematica

ODE 403

\(\sqrt {x^3+1} y'(x)=\sqrt {y(x)^3+1}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 404

\(\sqrt {(1-x) x (1-a x)} y'(x)=\sqrt {(1-y(x)) y(x) (1-a y(x))}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 405

\(\sqrt {1-x^4} y'(x)=\sqrt {1-y(x)^4}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 406

\(\sqrt {x^4+x^2+1} y'(x)=\sqrt {y(x)^4+y(x)^2+1}\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 407

\(\sqrt {X} y'(x)=0\)

[_quadrature]

Solution method Separable ODE, Dependent variable missing

Maple
Mathematica

ODE 408

\(\sqrt {X} y'(x)+\sqrt {Y}=0\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 409

\(\sqrt {X} y'(x)=\sqrt {Y}\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 410

\(\left (x^3+1\right )^{2/3} y'(x)+\left (y(x)^3+1\right )^{2/3}=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 411

\(\left (\text {a0}+\text {a1} x+4 x^3\right )^{2/3} y'(x)+\left (\text {a0}+\text {a1} y(x)+4 y(x)^3\right )^{2/3}=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 412

\(X^{2/3} y'(x)=Y^{2/3}\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 413

\(\left (a+\cos ^2\left (\frac {x}{2}\right )\right ) y'(x)=y(x) \tan \left (\frac {x}{2}\right ) \left (a-y(x)+\cos ^2\left (\frac {x}{2}\right )+1\right )\)

[_Bernoulli]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 414

\(\left (1-4 \cos ^2(x)\right ) y'(x)=y(x) \left (4 \cos ^2(x)+1\right ) \tan (x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 415

\((1-\sin (x)) y'(x)+y(x) \cos (x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 416

\(y'(x) (\cos (x)-\sin (x))+y(x) (\sin (x)+\cos (x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 417

\(y'(x) \left (\text {a0}+\text {a1} \sin ^2(x)\right )+\text {a2} x \left (\text {a1} \sin ^2(x)+\text {a3}\right )+\text {a1} y(x) \sin (2 x)=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 418

\(\left (x-e^x\right ) y'(x)+\left (1-e^x\right ) y(x)+e^x x=0\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 419

\(x \log (x) y'(x)=a x (\log (x)+1)-y(x)\)

[_linear]

Solution method Linear ODE

Maple
Mathematica

ODE 420

\(y(x) y'(x)+x=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 421

\(e^{x^2} x+y(x) y'(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 422

\(x^3+y(x) y'(x)+y(x)=0\)

[_rational, [_Abel, `2nd type`, `class A`]]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 423

\(a x+b y(x)+y(x) y'(x)=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 424

\(y(x) y'(x)+e^{-x} x (y(x)+1)=0\)

[_separable]

Solution method Linear ODE

Maple
Mathematica

ODE 425

\(f(x)+y(x) y'(x)=g(x) y(x)\)

[[_Abel, `2nd type`, `class A`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 426

\(y(x) y'(x)+y(x)^2+4 x (x+1)=0\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 427

\(y(x) y'(x)=a x+b y(x)^2\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 428

\(y(x) y'(x)=a y(x)^2+b \cos (c+x)\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 429

\(y(x) y'(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 430

\(y(x) y'(x)=a x+b x y(x)^2\)

[_separable]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 431

\(y(x) y'(x)=\csc ^2(x)-y(x)^2 \cot (x)\)

[_Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 432

\(y(x) y'(x)=\sqrt {a^2+y(x)^2}\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 433

\(y(x) y'(x)=\sqrt {y(x)^2-a^2}\)

[_quadrature]

Solution method Separable ODE, Independent variable missing

Maple
Mathematica

ODE 434

\(g(x) f\left (x^2+y(x)^2\right )+y(x) y'(x)+x=0\)

[NONE]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 435

\((y(x)+1) y'(x)=y(x)+x\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 436

\((y(x)+1) y'(x)=x^2 (1-y(x))\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 437

\((y(x)+x) y'(x)+y(x)=0\)

[[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 438

\((x-y(x)) y'(x)=y(x)\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 439

\((y(x)+x) y'(x)-y(x)+x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 440

\((y(x)+x) y'(x)=x-y(x)\)

[[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 441

\(1-y'(x)=y(x)+x\)

[[_linear, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 442

\((x-y(x)) y'(x)=y(x) (2 x y(x)+1)\)

[[_homogeneous, `class D`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 443

\((y(x)+x) y'(x)+\tan (y(x))=0\)

[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 444

\((x-y(x)) y'(x)=\left (e^{-\frac {x}{y(x)}}+1\right ) y(x)\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 445

\((y(x)+x+1) y'(x)+3 y(x)+4 x+1=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 446

\((y(x)+x+2) y'(x)=-y(x)-x+1\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 447

\((-y(x)-x+3) y'(x)=-3 y(x)+x+1\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 448

\((y(x)-x+3) y'(x)=3 y(x)-4 x+11\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 449

\((y(x)+2 x) y'(x)-2 y(x)+x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 450

\((-y(x)+2 x+2) y'(x)+3 (-y(x)+2 x+1)=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 451

\((-y(x)+2 x+3) y'(x)+2=0\)

[[_homogeneous, `class C`], [_Abel, `2nd type`, `class C`], _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 452

\((-y(x)+2 x+4) y'(x)-2 y(x)+x+5=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 453

\((-y(x)-2 x+5) y'(x)-2 y(x)-x+4=0\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 454

\((y(x)-3 x+1) y'(x)=2 (x-y(x))\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 455

\((y(x)-3 x+2) y'(x)-3 y(x)-2 x+5=0\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 456

\((4 x-y(x)) y'(x)-5 y(x)+2 x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 457

\((-y(x)-4 x+6) y'(x)=2 x-y(x)\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 458

\((-y(x)+5 x+1) y'(x)-5 y(x)+x+5=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 459

\(y'(x) (a+b x+y(x))+a-b x-y(x)=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 460

\(\left (x^2-y(x)\right ) y'(x)+x=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 461

\(\left (x^2-y(x)\right ) y'(x)=4 x y(x)\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 462

\(y'(x) (y(x)-\cot (x) \csc (x))+y(x) \csc (x) (y(x) \cos (x)+1)=0\)

[[_Abel, `2nd type`, `class A`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 463

\(x^2+2 y(x) y'(x)+y(x)^2+2 x=0\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 464

\(2 y(x) y'(x)=x^3+x y(x)^2\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 465

\((x-2 y(x)) y'(x)=y(x)\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 466

\((2 y(x)+x) y'(x)-y(x)+2 x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 467

\((x-2 y(x)) y'(x)+y(x)+2 x=0\)

[[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Exact equation

Maple
Mathematica

ODE 468

\((-2 y(x)+x+1) y'(x)=-y(x)+2 x+1\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Exact equation

Maple
Mathematica

ODE 469

\((2 y(x)+x+1) y'(x)-2 y(x)-x+1=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 470

\((2 y(x)+x+1) y'(x)-4 y(x)+x+7=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 471

\(x^2+2 (y(x)+x) y'(x)+2 y(x)=0\)

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]

Solution method Exact equation

Maple
Mathematica

ODE 472

\((-2 y(x)+2 x+3) y'(x)=-2 y(x)+6 x+1\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 473

\((-2 y(x)-4 x+1) y'(x)+y(x)+2 x=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 474

\((6 x-2 y(x)) y'(x)=-y(x)+3 x+2\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 475

\((2 y(x)+9 x+19) y'(x)-6 y(x)-2 x+18=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 476

\(\left (x^3+2 y(x)\right ) y'(x)=3 x (2-x y(x))\)

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]

Solution method Exact equation

Maple
Mathematica

ODE 477

\(y'(x) (\tan (x) \sec (x)-2 y(x))+\sec (x) (2 y(x) \sin (x)+1)=0\)

[[_Abel, `2nd type`, `class A`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 478

\(\left (e^{-x} x-2 y(x)\right ) y'(x)=2 e^{-2 x} x-\left (-2 y(x)+e^{-x} x+e^{-x}\right ) y(x)\)

[[_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 479

\(3 y(x) y'(x)+5 \cot (x) \cos ^2(y(x)) \cot (y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 480

\(3 (2-y(x)) y'(x)+x y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 481

\((x-3 y(x)) y'(x)-y(x)+3 x+4=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 482

\((-3 y(x)-x+4) y'(x)-3 y(x)-x+3=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 483

\((3 y(x)+2 x+2) y'(x)=-3 y(x)-2 x+1\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 484

\((-3 y(x)-2 x+5) y'(x)-3 y(x)-2 x+1=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 485

\((-3 y(x)+9 x+1) y'(x)-y(x)+3 x+2=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 486

\((4 y(x)+x) y'(x)-y(x)+4 x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 487

\((4 y(x)+2 x+3) y'(x)=2 y(x)+x+1\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 488

\((-4 y(x)+2 x+5) y'(x)=-2 y(x)+x+3\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 489

\((-4 y(x)+3 x+5) y'(x)=-3 y(x)+7 x+2\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 490

\(4 (-y(x)-x+1) y'(x)-x+2=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 491

\((-4 y(x)-11 x+11) y'(x)=-25 y(x)-8 x+62\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 492

\((5 y(x)+3 x+6) y'(x)=7 y(x)+x+2\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 493

\((5 y(x)+7 x) y'(x)+8 y(x)+10 x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 494

\(\left (4 x^3+5 y(x)+x\right ) y'(x)+7 x^3+3 x^2 y(x)+4 y(x)=0\)

[_rational, [_Abel, `2nd type`, `class A`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 495

\((6 y(x)-x+5) y'(x)=4 y(x)-x+3\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 496

\(3 (2 y(x)+x) y'(x)=-2 y(x)-x+1\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 497

\((7 y(x)-3 x+3) y'(x)+3 y(x)-7 x+7=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 498

\((9 y(x)+x+1) y'(x)+5 y(x)+x+1=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 499

\((-12 y(x)+5 x+8) y'(x)=-5 y(x)+2 x+3\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 500

\((-16 y(x)+7 x+140) y'(x)+y(x)+8 x+25=0\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 501

\((21 y(x)+9 x+3) y'(x)=-5 y(x)+7 x+45\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 502

\(y'(x) (a x+b y(x))+x=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 503

\(y'(x) (a x+b y(x))+y(x)=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 504

\(y'(x) (a x+b y(x))+a y(x)+b x=0\)

[[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Exact equation

Maple
Mathematica

ODE 505

\(y'(x) (a x+b y(x))=a y(x)+b x\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 506

\(\text {a1}+y'(x) (\text {a2}+b x+\text {c2} y(x))+b y(x)+\text {b1} x=0\)

[[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 507

\(y'(x) (\text {a2}+\text {b2} y(x)+\text {c2} y(x))=\text {a1}+\text {b1} x+\text {c1} y(x)\)

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 508

\(x y(x) y'(x)+y(x)^2+1=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 509

\(x y(x) y'(x)=y(x)^2+x\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 510

\(x^2+x y(x) y'(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method Homogeneous equation

Maple
Mathematica

ODE 511

\(x^4+x y(x) y'(x)-y(x)^2=0\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 512

\(x y(x) y'(x)=a x^3 \cos (x)+y(x)^2\)

[[_homogeneous, `class D`], _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 513

\(x y(x) y'(x)=x^2-x y(x)+y(x)^2\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 514

\(2 x^2+x y(x) y'(x)-2 x y(x)-y(x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 515

\(x y(x) y'(x)=a+b y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 516

\(x y(x) y'(x)=a x^n+b y(x)^2\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 517

\(x y(x) y'(x)=\left (x^2+1\right ) \left (1-y(x)^2\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 518

\(x^2 \cot ^{-1}\left (\frac {y(x)}{x}\right )+x y(x) y'(x)-y(x)^2=0\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 519

\(x^2 e^{-\frac {2 y(x)}{x}}+x y(x) y'(x)-y(x)^2=0\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 520

\((x y(x)+1) y'(x)+y(x)^2=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 521

\(x (y(x)+1) y'(x)-(1-x) y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 522

\(x (1-y(x)) y'(x)+(x+1) y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 523

\(x (1-y(x)) y'(x)+(1-x) y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 524

\(a x+x (y(x)+2) y'(x)=0\)

[_quadrature]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 525

\((x (-y(x))+3 x+2) y'(x)+y(x)=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 526

\(x (y(x)+4) y'(x)=y(x)^2+2 y(x)+2 x\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation, special

Maple
Mathematica

ODE 527

\(x (a+y(x)) y'(x)+b x+c y(x)=0\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 528

\(x (a+y(x)) y'(x)=y(x) (A+B x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 529

\(x (y(x)+x) y'(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 530

\(x (x-y(x)) y'(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 531

\(x (y(x)+x) y'(x)=x^2+y(x)^2\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 532

\(2 x^2+x (x-y(x)) y'(x)+3 x y(x)-y(x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 533

\(x \sqrt {x^2-y(x)^2}+x (y(x)+x) y'(x)-y(x) (y(x)+x)=0\)

[[_homogeneous, `class A`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 534

\((a+x (y(x)+x)) y'(x)=b y(x) (y(x)+x)\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation, special

Maple
Mathematica

ODE 535

\(x (y(x)+2 x) y'(x)=x^2+x y(x)-y(x)^2\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 536

\(4 x^2+x (4 x-y(x)) y'(x)-6 x y(x)-y(x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 537

\(x \left (x^3+y(x)\right ) y'(x)=\left (x^3-y(x)\right ) y(x)\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 538

\(x \left (2 x^3+y(x)\right ) y'(x)=\left (2 x^3-y(x)\right ) y(x)\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 539

\(x \left (2 x^3+y(x)\right ) y'(x)=6 y(x)^2\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 540

\((1-x) y(x) y'(x)+x (1-y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 541

\((a+x) (b+x) y'(x)=x y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 542

\(-2 x^3+2 x y(x) y'(x)-y(x)^2+1=0\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 543

\(a+2 x y(x) y'(x)+y(x)^2=0\)

[_separable]

Solution method Exact equation

Maple
Mathematica

ODE 544

\(2 x y(x) y'(x)=a x+y(x)^2\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 545

\(x^2+2 x y(x) y'(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _exact, _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 546

\(2 x y(x) y'(x)=x^2+y(x)^2\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 547

\(2 x y(x) y'(x)=4 (2 x+1) x^2+y(x)^2\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 548

\(x^2 \left (a x^3+1\right )+2 x y(x) y'(x)=6 y(x)^2\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 549

\(3 x^2+(2 x y(x)-x+3) y'(x)+y(x)^2-y(x)=0\)

[_exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 550

\(x (x-2 y(x)) y'(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 551

\(x (2 y(x)+x) y'(x)+(2 x-y(x)) y(x)=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 552

\(x (x-2 y(x)) y'(x)+(2 x-y(x)) y(x)=0\)

[[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 553

\(x (-2 y(x)+x+1) y'(x)+y(x) (y(x)-2 x+1)=0\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 554

\(x (-2 y(x)-x+1) y'(x)+y(x) (y(x)+2 x+1)=0\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 555

\(2 x \left (2 x^2+y(x)\right ) y'(x)+y(x) \left (12 x^2+y(x)\right )=0\)

[[_homogeneous, `class G`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 556

\(-3 x^2+2 (x+1) y(x) y'(x)+y(x)^2+2 x=0\)

[_exact, _rational, _Bernoulli]

Solution method Exact equation

Maple
Mathematica

ODE 557

\(x (3 y(x)+2 x) y'(x)=y(x)^2\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 558

\(x (3 y(x)+2 x) y'(x)+3 (y(x)+x)^2=0\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 559

\(\left (x^2+6 x y(x)+3\right ) y'(x)+3 y(x)^2+2 x y(x)+2 x=0\)

[_exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 560

\(x^3+3 x (2 y(x)+x) y'(x)+3 y(x) (y(x)+2 x)=0\)

[_exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 561

\(a x y(x) y'(x)=x^2+y(x)^2\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method Homogeneous equation

Maple
Mathematica

ODE 562

\(a x y(x) y'(x)+x^2-y(x)^2=0\)

[[_homogeneous, `class A`], _rational, _Bernoulli]

Solution method Homogeneous equation

Maple
Mathematica

ODE 563

\(x y'(x) (a+b y(x))=c y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 564

\(x (x-a y(x)) y'(x)=y(x) (y(x)-a x)\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation

Maple
Mathematica

ODE 565

\(y'(x) (x (A x+B y(x))+\text {a0}+\text {a1} x+\text {a2} y(x))=y(x) (A x+B y(x))+\text {b0}+\text {b1} x+\text {b2} y(x)\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation, Jacobi equation

Maple
Mathematica

ODE 566

\(y'(x) (x (\text {a1}+\text {b2} x+\text {c2} y(x))+\text {a1}+\text {b1} x+\text {c1} y(x))=y(x) (\text {a2}+\text {b2} x+\text {c2} y(x))+\text {a3}+\text {b3} x+\text {c2} y(x)\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation, Jacobi equation

Maple
Mathematica

ODE 567

\(x y'(x) \left (a y(x)+x^n\right )+y(x)^2 (b+c y(x))=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 568

\(\left (1-x^2 y(x)\right ) y'(x)-x y(x)^2+1=0\)

[_exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 569

\(\left (1-x^2 y(x)\right ) y'(x)+x y(x)^2-1=0\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation, special

Maple
Mathematica

ODE 570

\(x (1-x y(x)) y'(x)+y(x) (x y(x)+1)=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 571

\(x (x y(x)+2) y'(x)=2 x^3-x y(x)^2-2 y(x)+3\)

[_exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 572

\(x (2-x y(x)) y'(x)-x (x y(x)+1) y(x)^2+2 y(x)=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class C`]]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 573

\(x (3-x y(x)) y'(x)=y(x) (x y(x)-1)\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 574

\(x^2 (1-y(x)) y'(x)+(1-x) y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 575

\(x^2 (1-y(x)) y'(x)+(x+1) y(x)^2=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 576

\(\left (x^2+1\right ) y(x) y'(x)+x \left (1-y(x)^2\right )=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 577

\(\left (1-x^2\right ) y(x) y'(x)+2 x^2+x y(x)^2=0\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 578

\(2 x^2 y(x) y'(x)=x^2 (2 x+1)-y(x)^2\)

[[_homogeneous, `class D`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 579

\(x (1-2 x y(x)) y'(x)+y(x) (2 x y(x)+1)=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 580

\(x (2 x y(x)+1) y'(x)+y(x) (3 x y(x)+2)=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Homogeneous equation, isobaric equation

Maple
Mathematica

ODE 581

\(y(x) \left (-x^2 y(x)^2+2 x y(x)+1\right )+x (2 x y(x)+1) y'(x)=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class C`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 582

\(x^2 (x-2 y(x)) y'(x)=2 x^3-4 x y(x)^2+y(x)^3\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 583

\(2 x (x+1) y(x) y'(x)=y(x)^2+1\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 584

\(3 x^2 y(x) y'(x)+2 x y(x)^2+1=0\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 585

\(x^2 (4 x-3 y(x)) y'(x)=y(x) \left (6 x^2-3 x y(x)+2 y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 586

\(\left (1-x^3 y(x)\right ) y'(x)=x^2 y(x)^2\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 587

\(a+2 x^3 y(x) y'(x)+3 x^2 y(x)^2=0\)

[[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

Solution method Exact equation

Maple
Mathematica

ODE 588

\(x \left (3-2 x^2 y(x)\right ) y'(x)=3 x^2 y(x)^2-3 y(x)+4 x\)

[_exact, _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation

Maple
Mathematica

ODE 589

\(x \left (2 x^2 y(x)+3\right ) y'(x)+y(x) \left (3 x^2 y(x)+4\right )=0\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 590

\(3 x^4+8 x^3 y(x) y'(x)-6 x^2 y(x)^2-y(x)^4=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 591

\(x y(x) \left (a+b x^2\right ) y'(x)=A+B y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 592

\(3 x^4 y(x) y'(x)=1-2 x^3 y(x)^2\)

[[_homogeneous, `class G`], _rational, _Bernoulli]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 593

\(x^7 y(x) y'(x)=5 x^3 y(x)+2 \left (x^2+1\right )\)

[_rational, [_Abel, `2nd type`, `class B`]]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 594

\(\sqrt {x^2+1} y(x) y'(x)+x \sqrt {y(x)^2+1}=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 595

\(\sqrt {x^2+1} (y(x)+1) y'(x)=y(x)^3\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 596

\(y'(x) (\text {g0}(x)+\text {g1}(x) y(x))=\text {f0}(x)+\text {f1}(x) y(x)+\text {f2}(x) y(x)^2+\text {f3}(x) y(x)^3\)

[[_Abel, `2nd type`, `class C`]]

Solution method Abel ODE, Second kind

Maple
Mathematica

ODE 597

\(y(x)^2 y'(x)+x (2-y(x))=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 598

\(y(x)^2 y'(x)=x \left (y(x)^2+1\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 599

\(\left (y(x)^2+x\right ) y'(x)+y(x)=a+b x\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 600

\(\left (x-y(x)^2\right ) y'(x)=x^2-y(x)\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 601

\(\left (x^2+y(x)^2\right ) y'(x)+x y(x)=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 602

\(\left (x^2+y(x)^2\right ) y'(x)=x y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 603

\(\left (x^2-y(x)^2\right ) y'(x)=2 x y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 604

\(\left (x^2-y(x)^2\right ) y'(x)+x (2 y(x)+x)=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 605

\(\left (x^2+y(x)^2\right ) y'(x)+2 x (y(x)+2 x)=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 606

\(\left (-x^2+y(x)^2+1\right ) y'(x)=x^2-y(x)^2+1\)

[[_1st_order, _with_linear_symmetries], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 607

\(\left (a^2+x^2+y(x)^2\right ) y'(x)+2 x y(x)=0\)

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation

Maple
Mathematica

ODE 608

\(\left (a^2+x^2+y(x)^2\right ) y'(x)+b^2+x^2+2 x y(x)=0\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 609

\(\left (x^2+y(x)^2+x\right ) y'(x)=y(x)\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 610

\(\left (3 x^2-y(x)^2\right ) y'(x)=2 x y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 611

\(\left (x^4+y(x)^2\right ) y'(x)=4 x^3 y(x)\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 612

\(y(x) (y(x)+1) y'(x)=x (x+1)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 613

\(\left (y(x)^2+2 y(x)+x\right ) y'(x)+y(x)^2 (y(x)+x)^2+y(x) (y(x)+1)=0\)

[_rational]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 614

\(\left (x^2+y(x)^2+2 y(x)\right ) y'(x)+2 x=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 615

\(\left (x^3-y(x)^2+2 y(x)\right ) y'(x)+3 x^2 y(x)=0\)

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation

Maple
Mathematica

ODE 616

\(\left (y(x)^2+x y(x)+y(x)+1\right ) y'(x)+y(x)+1=0\)

[[_1st_order, _with_linear_symmetries], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 617

\((y(x)+x)^2 y'(x)=a^2\)

[[_homogeneous, `class C`], _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Maple
Mathematica

ODE 618

\((x-y(x))^2 y'(x)=a^2\)

[[_homogeneous, `class C`], _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Maple
Mathematica

ODE 619

\(\left (x^2+2 x y(x)-y(x)^2\right ) y'(x)+x^2-2 x y(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 620

\((x-y(x))^2 y'(x)=(-y(x)+x+1)^2\)

[[_homogeneous, `class C`], _rational, _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 621

\((y(x)+x)^2 y'(x)=(y(x)+x+2)^2\)

[[_homogeneous, `class C`], _rational, _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 622

\((y(x)+x)^2 y'(x)=x^2-2 x y(x)+5 y(x)^2\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 623

\(y'(x) (a+b+y(x)+x)^2=2 (a+y(x))^2\)

[[_homogeneous, `class C`], _rational]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 624

\(\left (2 x^2+4 x y(x)-y(x)^2\right ) y'(x)=x^2-4 x y(x)-2 y(x)^2\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 625

\((y(x)+3 x)^2 y'(x)=4 y(x) (2 y(x)+3 x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 626

\((-y(x)-3 x+1)^2 y'(x)=(-4 y(x)-6 x+3) (1-2 y(x))\)

[[_homogeneous, `class C`], _rational]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 627

\(y'(x) \left (\cot (x)-2 y(x)^2\right )=y(x)^3 \csc (x) \sec (x)\)

[`y=_G(x,y')`]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 628

\(3 y(x)^2 y'(x)=a y(x)^3+x+1\)

[_rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 629

\(\left (x^2-3 y(x)^2\right ) y'(x)+2 x y(x)+1=0\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 630

\(\left (2 x^2+3 y(x)^2\right ) y'(x)+x (y(x)+3 x)=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 631

\(3 \left (x^2-y(x)^2\right ) y'(x)-2 y(x)^3+6 x (x+1) y(x)+3 e^x=0\)

[`y=_G(x,y')`]

Solution method Change of Variable, new dependent variable

Maple
Mathematica

ODE 632

\(\left (3 x^2+2 x y(x)+4 y(x)^2\right ) y'(x)+2 x^2+6 x y(x)+y(x)^2=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 633

\((2 y(x)-3 x+1)^2 y'(x)=(-3 y(x)+2 x+4)^2\)

[[_homogeneous, `class C`], _rational]

Solution method Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)

Maple
Mathematica

ODE 634

\(\left (-3 x^2 y(x)+6 y(x)^2+1\right ) y'(x)+x^2-3 x y(x)^2=0\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 635

\(a+(x-6 y(x))^2 y'(x)-6 y(x)^2+2 x y(x)=0\)

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]

Solution method Exact equation

Maple
Mathematica

ODE 636

\(\left (a y(x)^2+x^2\right ) y'(x)=x y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 637

\(\left (a y(x)^2+x^2+x y(x)\right ) y'(x)=a x^2+x y(x)+y(x)^2\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 638

\(\left (a x^2-a y(x)^2+2 x y(x)\right ) y'(x)-2 a x y(x)+x^2-y(x)^2=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 639

\(y'(x) \left (x^2 (a+2 b)+2 x (2 b+c) y(x)+3 c y(x)^2\right )+2 x (a+2 b) y(x)+3 a x^2+(2 b+c) y(x)^2=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 640

\(y'(x) \left (a x^2+2 b x y(x)+c y(x)^2\right )+2 a x y(x)+b y(x)^2+k x^2=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 641

\(x \left (1-y(x)^2\right ) y'(x)=\left (x^2+1\right ) y(x)\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 642

\(x \left (3 x-y(x)^2\right ) y'(x)+y(x) \left (5 x-2 y(x)^2\right )=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 643

\(x \left (x^2+y(x)^2\right ) y'(x)=y(x) \left (x^4+x^2+y(x)^2\right )\)

[[_homogeneous, `class D`], _rational]

Solution method Homogeneous equation, \(xy'(x)=x f(x) g(u)+y(x)\)

Maple
Mathematica

ODE 644

\(x \left (-x^2+y(x)^2+1\right ) y'(x)+y(x) \left (x^2-y(x)^2+1\right )=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Homogeneous equation, special

Maple
Mathematica

ODE 645

\(x \left (a-x^2-y(x)^2\right ) y'(x)+y(x) \left (a+x^2+y(x)^2\right )=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Solution method Homogeneous equation, special

Maple
Mathematica

ODE 646

\(x \left (2 x^2+y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 647

\(\left (x \left (a-x^2-y(x)^2\right )+y(x)\right ) y'(x)-y(x) \left (a-x^2-y(x)^2\right )+x=0\)

[[_1st_order, _with_linear_symmetries], _rational]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 648

\(x (a+y(x))^2 y'(x)=b y(x)^2\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 649

\(x \left (x^2-x y(x)+y(x)^2\right ) y'(x)+y(x) \left (x^2+x y(x)+y(x)^2\right )=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 650

\(x \left (x^2-x y(x)-y(x)^2\right ) y'(x)=y(x) \left (x^2+x y(x)-y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 651

\(x \left (a x y(x)+x^2+y(x)^2\right ) y'(x)=y(x) \left (b x y(x)+x^2+y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 652

\(x \left (x^2-2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2-y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 653

\(x \left (x^2+2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 654

\(2 x \left (5 x^2+y(x)^2\right ) y'(x)=x^2 y(x)-y(x)^3\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 655

\(x \left (a x y(x)+x^2+2 y(x)^2\right ) y'(x)=y(x)^2 (a x+2 y(x))\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 656

\(3 x y(x)^2 y'(x)=2 x-y(x)^3\)

[[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 657

\(\left (3 x y(x)^2-4 x+1\right ) y'(x)=y(x) \left (2-y(x)^2\right )\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 658

\(x \left (x-3 y(x)^2\right ) y'(x)+y(x) \left (2 x-y(x)^2\right )=0\)

[[_homogeneous, `class G`], _exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 659

\(x^3+3 x \left (y(x)^2+x\right ) y'(x)-3 x y(x)-2 y(x)^3=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 660

\(x \left (-3 x^3 y(x)+x^3+4 y(x)^2\right ) y'(x)=6 y(x)^3\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 661

\(6 x y(x)^2 y'(x)+2 y(x)^3+x=0\)

[[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

Solution method The Bernoulli ODE

Maple
Mathematica

ODE 662

\(x \left (6 y(x)^2+x\right ) y'(x)-3 y(x)^3+x y(x)=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 663

\(x \left (x^2-6 y(x)^2\right ) y'(x)=4 y(x) \left (x^2+3 y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 664

\(x \left (3 x-7 y(x)^2\right ) y'(x)+y(x) \left (5 x-3 y(x)^2\right )=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 665

\(x^3+x^2 y(x)^2 y'(x)-x+1=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 666

\(\left (1-x^2 y(x)^2\right ) y'(x)=x y(x)^3\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 667

\(\left (1-x^2 y(x)^2\right ) y'(x)=y(x)^2 (x y(x)+1)\)

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 668

\(x \left (x y(x)^2+1\right ) y'(x)+y(x)=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 669

\(x \left (x y(x)^2+1\right ) y'(x)=y(x) \left (2-3 x y(x)^2\right )\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 670

\(x^2 (a+y(x))^2 y'(x)=\left (x^2+1\right ) \left (a^2+y(x)^2\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 671

\(\left (x^2+1\right ) \left (y(x)^2+1\right ) y'(x)+2 x y(x) \left (1-y(x)^2\right )=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 672

\(\left (x^2+1\right ) \left (y(x)^2+1\right ) y'(x)+2 x y(x) (1-y(x))^2=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 673

\(\left (-x^3+6 x^2 y(x)^2+1\right ) y'(x)=x \left (-4 y(x)^3+3 x y(x)+6\right )\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 674

\(x \left (4 x^2 y(x)-12 x y(x)^2+5 x+3\right ) y'(x)+y(x) \left (6 x^2 y(x)-8 x y(x)^2+10 x+3\right )=0\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 675

\(x^3 \left (y(x)^2+1\right ) y'(x)+3 x^2 y(x)=0\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 676

\(y(x) \left (x^2 y(x)^2+1\right )+x (1-x y(x))^2 y'(x)=0\)

[[_homogeneous, `class G`], _rational]

Solution method Homogeneous equation, isobaric equation

Maple
Mathematica

ODE 677

\(\left (1-x^4 y(x)^2\right ) y'(x)=x^3 y(x)^3\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 678

\(\left (3 x-y(x)^3\right ) y'(x)=x^2-3 y(x)\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 679

\(\left (x^3-y(x)^3\right ) y'(x)+x^2 y(x)=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 680

\(x^2 (a x+3 y(x))+\left (x^3+y(x)^3\right ) y'(x)=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 681

\(\left (-x^2 y(x)-y(x)^3+x\right ) y'(x)=x^3+x y(x)^2-y(x)\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 682

\(\left (a^2 x+y(x) \left (x^2-y(x)^2\right )\right ) y'(x)+x \left (x^2-y(x)^2\right )=a^2 y(x)\)

[_rational]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 683

\(y(x) \left (a+x^2+y(x)^2\right ) y'(x)=x \left (a-x^2-y(x)^2\right )\)

[_exact, _rational]

Solution method Exact equation

Maple
Mathematica

ODE 684

\(y(x) \left (3 x^2+y(x)^2\right ) y'(x)+x \left (x^2+3 y(x)^2\right )=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 685

\(y(x) \left (a-3 x^2-y(x)^2\right ) y'(x)+x \left (a-x^2+y(x)^2\right )=0\)

[_rational]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 686

\(2 y(x)^3 y'(x)=x^3-x y(x)^2\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 687

\(y(x) \left (2 y(x)^2+1\right ) y'(x)=x \left (2 x^2+1\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 688

\(x^3+y(x) \left (3 x^2+2 y(x)^2\right ) y'(x)=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 689

\(y(x) \left (5 x^2+2 y(x)^2\right ) y'(x)+x \left (x^2+5 y(x)^2\right )=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 690

\(2 x^3+3 x^2 y(x)+\left (-x^3+x^2+3 x y(x)^2+2 y(x)^3\right ) y'(x)-y(x)^3+y(x)^2=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 691

\(4 x^3+9 x^2 y(x)+\left (3 x^3+6 x^2 y(x)-3 x y(x)^2+20 y(x)^3\right ) y'(x)+6 x y(x)^2-y(x)^3=0\)

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Solution method Exact equation

Maple
Mathematica

ODE 692

\(\left (a y(x)^3+x^3\right ) y'(x)=x^2 y(x)\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 693

\(y'(x) \left (a x^2-b x^3+3 c x y(x)^2+2 c y(x)^3\right )+a y(x)^2+2 b x^3+3 b x^2 y(x)-c y(x)^3=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 694

\(x y(x)^3 y'(x)=\left (1-x^2\right ) \left (y(x)^2+1\right )\)

[_separable]

Solution method Separable ODE, Neither variable missing

Maple
Mathematica

ODE 695

\(x \left (x-y(x)^3\right ) y'(x)=y(x) \left (y(x)^3+3 x\right )\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 696

\(x \left (2 x^3+y(x)^3\right ) y'(x)=y(x) \left (2 x^3-x^2 y(x)+y(x)^3\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 697

\(x \left (2 x^3-y(x)^3\right ) y'(x)=y(x) \left (x^3-2 y(x)^3\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 698

\(x \left (x^3+3 x^2 y(x)+y(x)^3\right ) y'(x)=y(x)^2 \left (3 x^2+y(x)^2\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Homogeneous equation

Maple
Mathematica

ODE 699

\(x \left (x^3-2 y(x)^3\right ) y'(x)=y(x) \left (2 x^3-y(x)^3\right )\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 700

\(x \left (x^4-2 y(x)^3\right ) y'(x)+y(x) \left (2 x^4+y(x)^3\right )=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 701

\(x \left (2 y(x)^3+y(x)+x\right ) y'(x)=(x-y(x)) y(x)\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 702

\(\left (-7 x y(x)^3-y(x)+5 x\right ) y'(x)-y(x)^4+5 y(x)=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 703

\(y(x) \left (1-2 x^3 y(x)\right )+x \left (1-2 x y(x)^3\right ) y'(x)=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 704

\(x \left (-2 x y(x)^3-x y(x)^2+2\right ) y'(x)+2 y(x)+1=0\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 705

\(\left (-10 x^2 y(x)^3+3 y(x)^2+2\right ) y'(x)=x \left (5 y(x)^4+1\right )\)

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation

Maple
Mathematica

ODE 706

\(x y'(x) \left (a+b x y(x)^3\right )+y(x) \left (a+c x^3 y(x)\right )=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 707

\(y(x) \left (1-2 x^3 y(x)^2\right )+x \left (1-2 x^2 y(x)^3\right ) y'(x)=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 708

\(x (1-x y(x)) \left (1-x^2 y(x)^2\right ) y'(x)+y(x) (x y(x)+1) \left (x^2 y(x)^2+1\right )=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 709

\(\left (x^2-y(x)^4\right ) y'(x)=x y(x)\)

[[_homogeneous, `class G`], _rational]

Solution method Change of Variable, new independent variable

Maple
Mathematica

ODE 710

\(\left (x^3-y(x)^4\right ) y'(x)=3 x^2 y(x)\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 711

\(\left (a^2 x^2+\left (x^2+y(x)^2\right )^2\right ) y'(x)=a^2 x y(x)\)

[_rational]

Solution method Change of Variable, Two new variables

Maple
Mathematica

ODE 712

\(2 \left (x-y(x)^4\right ) y'(x)=y(x)\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 713

\(\left (-2 y(x)^4-x y(x)^3+4 x\right ) y'(x)=y(x) \left (y(x)^3+2\right )\)

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 714

\(y(x) y'(x) \left ((a x+b y(x))^3+a x^3\right )+x \left ((a x+b y(x))^3+b y(x)^3\right )=0\)

[[_homogeneous, `class A`], _rational, _dAlembert]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 715

\(\left (2 x^2 y(x)^3+x y(x)^4+2 y(x)+x\right ) y'(x)+y(x) \left (y(x)^4+1\right )=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 716

\(2 x \left (x^3+y(x)^4\right ) y'(x)=y(x) \left (x^3+2 y(x)^4\right )\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 717

\(x \left (1-x^2 y(x)^4\right ) y'(x)+y(x)=0\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 718

\(\left (x^2-y(x)^5\right ) y'(x)=2 x y(x)\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 719

\(x \left (x^3+y(x)^5\right ) y'(x)=y(x) \left (x^3-y(x)^5\right )\)

[[_homogeneous, `class G`], _rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 720

\(y(x)^3 \left (3 x^5 y(x)^5-1\right )+x^3 \left (5 x^3 y(x)^7+1\right ) y'(x)=0\)

[_rational]

Solution method Exact equation, integrating factor

Maple
Mathematica

ODE 721

\(y'(x) (\text {f1}(x,y(x))+x \text {f2}(x,y(x)))=y(x) \text {f2}(x,y(x))+\text {f3}(x,y(x))\)

[NONE]

Solution method Homogeneous equation, Darbooux type

Maple
Mathematica

ODE 722

\(y'(x) (a (y(x)+x)+1)^n+a (y(x)+x)^n=0\)

[[_homogeneous, `class C`], _dAlembert]

Solution method Exact equation

Maple