4.10.41 \((-4 y(x)-11 x+11) y'(x)=-25 y(x)-8 x+62\)

ODE
\[ (-4 y(x)-11 x+11) y'(x)=-25 y(x)-8 x+62 \] ODE Classification

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

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

Mathematica ✓
cpu = 0.427065 (sec), leaf count = 1677

\[\left \{\left \{y(x)\to \frac {1}{4} \left (\frac {3 (9 x-1)}{\frac {\sqrt [3]{-6561 \cosh \left (\frac {3 c_1}{4}\right ) x^4-6561 \sinh \left (\frac {3 c_1}{4}\right ) x^4+2916 \cosh \left (\frac {3 c_1}{4}\right ) x^3+2916 \sinh \left (\frac {3 c_1}{4}\right ) x^3+162 \cosh \left (\frac {3 c_1}{8}\right ) x^2-486 \cosh \left (\frac {3 c_1}{4}\right ) x^2+162 \sinh \left (\frac {3 c_1}{8}\right ) x^2-486 \sinh \left (\frac {3 c_1}{4}\right ) x^2-36 \cosh \left (\frac {3 c_1}{8}\right ) x+36 \cosh \left (\frac {3 c_1}{4}\right ) x-36 \sinh \left (\frac {3 c_1}{8}\right ) x+36 \sinh \left (\frac {3 c_1}{4}\right ) x+2 \cosh \left (\frac {3 c_1}{8}\right )-\cosh \left (\frac {3 c_1}{4}\right )+2 \sinh \left (\frac {3 c_1}{8}\right )-\sinh \left (\frac {3 c_1}{4}\right )+\sqrt {(1-9 x)^2 \left (9 x (9 x-2) \cosh \left (\frac {3 c_1}{16}\right )+\left (81 x^2-18 x+2\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right )}-1}}{\cosh \left (\frac {3 c_1}{8}\right ) (1-9 x)^2+\sinh \left (\frac {3 c_1}{8}\right ) (1-9 x)^2-1}+1-\frac {1}{\sqrt [3]{-6561 \cosh \left (\frac {3 c_1}{4}\right ) x^4-6561 \sinh \left (\frac {3 c_1}{4}\right ) x^4+2916 \cosh \left (\frac {3 c_1}{4}\right ) x^3+2916 \sinh \left (\frac {3 c_1}{4}\right ) x^3+162 \cosh \left (\frac {3 c_1}{8}\right ) x^2-486 \cosh \left (\frac {3 c_1}{4}\right ) x^2+162 \sinh \left (\frac {3 c_1}{8}\right ) x^2-486 \sinh \left (\frac {3 c_1}{4}\right ) x^2-36 \cosh \left (\frac {3 c_1}{8}\right ) x+36 \cosh \left (\frac {3 c_1}{4}\right ) x-36 \sinh \left (\frac {3 c_1}{8}\right ) x+36 \sinh \left (\frac {3 c_1}{4}\right ) x+2 \cosh \left (\frac {3 c_1}{8}\right )-\cosh \left (\frac {3 c_1}{4}\right )+2 \sinh \left (\frac {3 c_1}{8}\right )-\sinh \left (\frac {3 c_1}{4}\right )+\sqrt {(1-9 x)^2 \left (9 x (9 x-2) \cosh \left (\frac {3 c_1}{16}\right )+\left (81 x^2-18 x+2\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right )}-1}}}-11 (x-1)\right )\right \},\left \{y(x)\to \frac {1}{4} \left (\frac {6 (9 x-1)}{\frac {i \left (i+\sqrt {3}\right ) \sqrt [3]{-6561 \cosh \left (\frac {3 c_1}{4}\right ) x^4-6561 \sinh \left (\frac {3 c_1}{4}\right ) x^4+2916 \cosh \left (\frac {3 c_1}{4}\right ) x^3+2916 \sinh \left (\frac {3 c_1}{4}\right ) x^3+162 \cosh \left (\frac {3 c_1}{8}\right ) x^2-486 \cosh \left (\frac {3 c_1}{4}\right ) x^2+162 \sinh \left (\frac {3 c_1}{8}\right ) x^2-486 \sinh \left (\frac {3 c_1}{4}\right ) x^2-36 \cosh \left (\frac {3 c_1}{8}\right ) x+36 \cosh \left (\frac {3 c_1}{4}\right ) x-36 \sinh \left (\frac {3 c_1}{8}\right ) x+36 \sinh \left (\frac {3 c_1}{4}\right ) x+2 \cosh \left (\frac {3 c_1}{8}\right )-\cosh \left (\frac {3 c_1}{4}\right )+2 \sinh \left (\frac {3 c_1}{8}\right )-\sinh \left (\frac {3 c_1}{4}\right )+\sqrt {(1-9 x)^2 \left (9 x (9 x-2) \cosh \left (\frac {3 c_1}{16}\right )+\left (81 x^2-18 x+2\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right )}-1}}{\cosh \left (\frac {3 c_1}{8}\right ) (1-9 x)^2+\sinh \left (\frac {3 c_1}{8}\right ) (1-9 x)^2-1}+2+\frac {1+i \sqrt {3}}{\sqrt [3]{-6561 \cosh \left (\frac {3 c_1}{4}\right ) x^4-6561 \sinh \left (\frac {3 c_1}{4}\right ) x^4+2916 \cosh \left (\frac {3 c_1}{4}\right ) x^3+2916 \sinh \left (\frac {3 c_1}{4}\right ) x^3+162 \cosh \left (\frac {3 c_1}{8}\right ) x^2-486 \cosh \left (\frac {3 c_1}{4}\right ) x^2+162 \sinh \left (\frac {3 c_1}{8}\right ) x^2-486 \sinh \left (\frac {3 c_1}{4}\right ) x^2-36 \cosh \left (\frac {3 c_1}{8}\right ) x+36 \cosh \left (\frac {3 c_1}{4}\right ) x-36 \sinh \left (\frac {3 c_1}{8}\right ) x+36 \sinh \left (\frac {3 c_1}{4}\right ) x+2 \cosh \left (\frac {3 c_1}{8}\right )-\cosh \left (\frac {3 c_1}{4}\right )+2 \sinh \left (\frac {3 c_1}{8}\right )-\sinh \left (\frac {3 c_1}{4}\right )+\sqrt {(1-9 x)^2 \left (9 x (9 x-2) \cosh \left (\frac {3 c_1}{16}\right )+\left (81 x^2-18 x+2\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right )}-1}}}-11 (x-1)\right )\right \},\left \{y(x)\to \frac {1}{4} \left (\frac {6 (9 x-1)}{-\frac {i \left (-i+\sqrt {3}\right ) \sqrt [3]{-6561 \cosh \left (\frac {3 c_1}{4}\right ) x^4-6561 \sinh \left (\frac {3 c_1}{4}\right ) x^4+2916 \cosh \left (\frac {3 c_1}{4}\right ) x^3+2916 \sinh \left (\frac {3 c_1}{4}\right ) x^3+162 \cosh \left (\frac {3 c_1}{8}\right ) x^2-486 \cosh \left (\frac {3 c_1}{4}\right ) x^2+162 \sinh \left (\frac {3 c_1}{8}\right ) x^2-486 \sinh \left (\frac {3 c_1}{4}\right ) x^2-36 \cosh \left (\frac {3 c_1}{8}\right ) x+36 \cosh \left (\frac {3 c_1}{4}\right ) x-36 \sinh \left (\frac {3 c_1}{8}\right ) x+36 \sinh \left (\frac {3 c_1}{4}\right ) x+2 \cosh \left (\frac {3 c_1}{8}\right )-\cosh \left (\frac {3 c_1}{4}\right )+2 \sinh \left (\frac {3 c_1}{8}\right )-\sinh \left (\frac {3 c_1}{4}\right )+\sqrt {(1-9 x)^2 \left (9 x (9 x-2) \cosh \left (\frac {3 c_1}{16}\right )+\left (81 x^2-18 x+2\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right )}-1}}{\cosh \left (\frac {3 c_1}{8}\right ) (1-9 x)^2+\sinh \left (\frac {3 c_1}{8}\right ) (1-9 x)^2-1}+2+\frac {1-i \sqrt {3}}{\sqrt [3]{-6561 \cosh \left (\frac {3 c_1}{4}\right ) x^4-6561 \sinh \left (\frac {3 c_1}{4}\right ) x^4+2916 \cosh \left (\frac {3 c_1}{4}\right ) x^3+2916 \sinh \left (\frac {3 c_1}{4}\right ) x^3+162 \cosh \left (\frac {3 c_1}{8}\right ) x^2-486 \cosh \left (\frac {3 c_1}{4}\right ) x^2+162 \sinh \left (\frac {3 c_1}{8}\right ) x^2-486 \sinh \left (\frac {3 c_1}{4}\right ) x^2-36 \cosh \left (\frac {3 c_1}{8}\right ) x+36 \cosh \left (\frac {3 c_1}{4}\right ) x-36 \sinh \left (\frac {3 c_1}{8}\right ) x+36 \sinh \left (\frac {3 c_1}{4}\right ) x+2 \cosh \left (\frac {3 c_1}{8}\right )-\cosh \left (\frac {3 c_1}{4}\right )+2 \sinh \left (\frac {3 c_1}{8}\right )-\sinh \left (\frac {3 c_1}{4}\right )+\sqrt {(1-9 x)^2 \left (9 x (9 x-2) \cosh \left (\frac {3 c_1}{16}\right )+\left (81 x^2-18 x+2\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right )}-1}}}-11 (x-1)\right )\right \}\right \}\]

Maple ✓
cpu = 0.236 (sec), leaf count = 377

\[\left [y \left (x \right ) = \frac {22}{9}+\frac {36 \left (-1+9 x \right ) \left (-\frac {\left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}{27}-\frac {16}{27 \left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}-\frac {19}{27}+2 i \sqrt {3}\, \left (\frac {\left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}{54}-\frac {8}{27 \left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}\right )\right )}{-3 \left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}-\frac {48}{\left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}+24+162 i \sqrt {3}\, \left (\frac {\left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}{54}-\frac {8}{27 \left (64-8748 \left (-1+9 x \right )^{2} \textit {\_C1} +108 \sqrt {6561 \left (-1+9 x \right )^{4} \textit {\_C1}^{2}-96 \left (-1+9 x \right )^{2} \textit {\_C1}}\right )^{\frac {1}{3}}}\right )}\right ]\] Mathematica raw input

DSolve[(11 - 11*x - 4*y[x])*y'[x] == 62 - 8*x - 25*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (-11*(-1 + x) + (3*(-1 + 9*x))/(1 - (-1 + 2*Cosh[(3*C[1])/8] - 36*x*Co
sh[(3*C[1])/8] + 162*x^2*Cosh[(3*C[1])/8] - Cosh[(3*C[1])/4] + 36*x*Cosh[(3*C[1]
)/4] - 486*x^2*Cosh[(3*C[1])/4] + 2916*x^3*Cosh[(3*C[1])/4] - 6561*x^4*Cosh[(3*C
[1])/4] + 2*Sinh[(3*C[1])/8] - 36*x*Sinh[(3*C[1])/8] + 162*x^2*Sinh[(3*C[1])/8] 
- Sinh[(3*C[1])/4] + 36*x*Sinh[(3*C[1])/4] - 486*x^2*Sinh[(3*C[1])/4] + 2916*x^3
*Sinh[(3*C[1])/4] - 6561*x^4*Sinh[(3*C[1])/4] + Sqrt[(1 - 9*x)^2*(9*x*(-2 + 9*x)
*Cosh[(3*C[1])/16] + (2 - 18*x + 81*x^2)*Sinh[(3*C[1])/16])^3*(Cosh[(15*C[1])/16
] + Sinh[(15*C[1])/16])])^(-1/3) + (-1 + 2*Cosh[(3*C[1])/8] - 36*x*Cosh[(3*C[1])
/8] + 162*x^2*Cosh[(3*C[1])/8] - Cosh[(3*C[1])/4] + 36*x*Cosh[(3*C[1])/4] - 486*
x^2*Cosh[(3*C[1])/4] + 2916*x^3*Cosh[(3*C[1])/4] - 6561*x^4*Cosh[(3*C[1])/4] + 2
*Sinh[(3*C[1])/8] - 36*x*Sinh[(3*C[1])/8] + 162*x^2*Sinh[(3*C[1])/8] - Sinh[(3*C
[1])/4] + 36*x*Sinh[(3*C[1])/4] - 486*x^2*Sinh[(3*C[1])/4] + 2916*x^3*Sinh[(3*C[
1])/4] - 6561*x^4*Sinh[(3*C[1])/4] + Sqrt[(1 - 9*x)^2*(9*x*(-2 + 9*x)*Cosh[(3*C[
1])/16] + (2 - 18*x + 81*x^2)*Sinh[(3*C[1])/16])^3*(Cosh[(15*C[1])/16] + Sinh[(1
5*C[1])/16])])^(1/3)/(-1 + (1 - 9*x)^2*Cosh[(3*C[1])/8] + (1 - 9*x)^2*Sinh[(3*C[
1])/8])))/4}, {y[x] -> (-11*(-1 + x) + (6*(-1 + 9*x))/(2 + (1 + I*Sqrt[3])/(-1 +
 2*Cosh[(3*C[1])/8] - 36*x*Cosh[(3*C[1])/8] + 162*x^2*Cosh[(3*C[1])/8] - Cosh[(3
*C[1])/4] + 36*x*Cosh[(3*C[1])/4] - 486*x^2*Cosh[(3*C[1])/4] + 2916*x^3*Cosh[(3*
C[1])/4] - 6561*x^4*Cosh[(3*C[1])/4] + 2*Sinh[(3*C[1])/8] - 36*x*Sinh[(3*C[1])/8
] + 162*x^2*Sinh[(3*C[1])/8] - Sinh[(3*C[1])/4] + 36*x*Sinh[(3*C[1])/4] - 486*x^
2*Sinh[(3*C[1])/4] + 2916*x^3*Sinh[(3*C[1])/4] - 6561*x^4*Sinh[(3*C[1])/4] + Sqr
t[(1 - 9*x)^2*(9*x*(-2 + 9*x)*Cosh[(3*C[1])/16] + (2 - 18*x + 81*x^2)*Sinh[(3*C[
1])/16])^3*(Cosh[(15*C[1])/16] + Sinh[(15*C[1])/16])])^(1/3) + (I*(I + Sqrt[3])*
(-1 + 2*Cosh[(3*C[1])/8] - 36*x*Cosh[(3*C[1])/8] + 162*x^2*Cosh[(3*C[1])/8] - Co
sh[(3*C[1])/4] + 36*x*Cosh[(3*C[1])/4] - 486*x^2*Cosh[(3*C[1])/4] + 2916*x^3*Cos
h[(3*C[1])/4] - 6561*x^4*Cosh[(3*C[1])/4] + 2*Sinh[(3*C[1])/8] - 36*x*Sinh[(3*C[
1])/8] + 162*x^2*Sinh[(3*C[1])/8] - Sinh[(3*C[1])/4] + 36*x*Sinh[(3*C[1])/4] - 4
86*x^2*Sinh[(3*C[1])/4] + 2916*x^3*Sinh[(3*C[1])/4] - 6561*x^4*Sinh[(3*C[1])/4] 
+ Sqrt[(1 - 9*x)^2*(9*x*(-2 + 9*x)*Cosh[(3*C[1])/16] + (2 - 18*x + 81*x^2)*Sinh[
(3*C[1])/16])^3*(Cosh[(15*C[1])/16] + Sinh[(15*C[1])/16])])^(1/3))/(-1 + (1 - 9*
x)^2*Cosh[(3*C[1])/8] + (1 - 9*x)^2*Sinh[(3*C[1])/8])))/4}, {y[x] -> (-11*(-1 + 
x) + (6*(-1 + 9*x))/(2 + (1 - I*Sqrt[3])/(-1 + 2*Cosh[(3*C[1])/8] - 36*x*Cosh[(3
*C[1])/8] + 162*x^2*Cosh[(3*C[1])/8] - Cosh[(3*C[1])/4] + 36*x*Cosh[(3*C[1])/4] 
- 486*x^2*Cosh[(3*C[1])/4] + 2916*x^3*Cosh[(3*C[1])/4] - 6561*x^4*Cosh[(3*C[1])/
4] + 2*Sinh[(3*C[1])/8] - 36*x*Sinh[(3*C[1])/8] + 162*x^2*Sinh[(3*C[1])/8] - Sin
h[(3*C[1])/4] + 36*x*Sinh[(3*C[1])/4] - 486*x^2*Sinh[(3*C[1])/4] + 2916*x^3*Sinh
[(3*C[1])/4] - 6561*x^4*Sinh[(3*C[1])/4] + Sqrt[(1 - 9*x)^2*(9*x*(-2 + 9*x)*Cosh
[(3*C[1])/16] + (2 - 18*x + 81*x^2)*Sinh[(3*C[1])/16])^3*(Cosh[(15*C[1])/16] + S
inh[(15*C[1])/16])])^(1/3) - (I*(-I + Sqrt[3])*(-1 + 2*Cosh[(3*C[1])/8] - 36*x*C
osh[(3*C[1])/8] + 162*x^2*Cosh[(3*C[1])/8] - Cosh[(3*C[1])/4] + 36*x*Cosh[(3*C[1
])/4] - 486*x^2*Cosh[(3*C[1])/4] + 2916*x^3*Cosh[(3*C[1])/4] - 6561*x^4*Cosh[(3*
C[1])/4] + 2*Sinh[(3*C[1])/8] - 36*x*Sinh[(3*C[1])/8] + 162*x^2*Sinh[(3*C[1])/8]
 - Sinh[(3*C[1])/4] + 36*x*Sinh[(3*C[1])/4] - 486*x^2*Sinh[(3*C[1])/4] + 2916*x^
3*Sinh[(3*C[1])/4] - 6561*x^4*Sinh[(3*C[1])/4] + Sqrt[(1 - 9*x)^2*(9*x*(-2 + 9*x
)*Cosh[(3*C[1])/16] + (2 - 18*x + 81*x^2)*Sinh[(3*C[1])/16])^3*(Cosh[(15*C[1])/1
6] + Sinh[(15*C[1])/16])])^(1/3))/(-1 + (1 - 9*x)^2*Cosh[(3*C[1])/8] + (1 - 9*x)
^2*Sinh[(3*C[1])/8])))/4}}

Maple raw input

dsolve((11-11*x-4*y(x))*diff(y(x),x) = 62-8*x-25*y(x), y(x))

Maple raw output

[y(x) = 22/9+36*(-1+9*x)*(-1/27*(64-8748*(-1+9*x)^2*_C1+108*(6561*(-1+9*x)^4*_C1
^2-96*(-1+9*x)^2*_C1)^(1/2))^(1/3)-16/27/(64-8748*(-1+9*x)^2*_C1+108*(6561*(-1+9
*x)^4*_C1^2-96*(-1+9*x)^2*_C1)^(1/2))^(1/3)-19/27+2*I*3^(1/2)*(1/54*(64-8748*(-1
+9*x)^2*_C1+108*(6561*(-1+9*x)^4*_C1^2-96*(-1+9*x)^2*_C1)^(1/2))^(1/3)-8/27/(64-
8748*(-1+9*x)^2*_C1+108*(6561*(-1+9*x)^4*_C1^2-96*(-1+9*x)^2*_C1)^(1/2))^(1/3)))
/(-3*(64-8748*(-1+9*x)^2*_C1+108*(6561*(-1+9*x)^4*_C1^2-96*(-1+9*x)^2*_C1)^(1/2)
)^(1/3)-48/(64-8748*(-1+9*x)^2*_C1+108*(6561*(-1+9*x)^4*_C1^2-96*(-1+9*x)^2*_C1)
^(1/2))^(1/3)+24+162*I*3^(1/2)*(1/54*(64-8748*(-1+9*x)^2*_C1+108*(6561*(-1+9*x)^
4*_C1^2-96*(-1+9*x)^2*_C1)^(1/2))^(1/3)-8/27/(64-8748*(-1+9*x)^2*_C1+108*(6561*(
-1+9*x)^4*_C1^2-96*(-1+9*x)^2*_C1)^(1/2))^(1/3)))]