##### 4.15.2 $$\left (-7 x y(x)^3-y(x)+5 x\right ) y'(x)-y(x)^4+5 y(x)=0$$

ODE
$\left (-7 x y(x)^3-y(x)+5 x\right ) y'(x)-y(x)^4+5 y(x)=0$ ODE Classiﬁcation

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.370736 (sec), leaf count = 302

$\left \{\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,6\right ]\right \},\left \{y(x)\to \text {Root}\left [10 \text {\#1}^7 x+2 \text {\#1}^5-100 \text {\#1}^4 x-25 \text {\#1}^2+250 \text {\#1} x-10 c_1\& ,7\right ]\right \}\right \}$

Maple
cpu = 0.037 (sec), leaf count = 33

$\left [x -\frac {-\frac {y \left (x \right )^{5}}{5}+\frac {5 y \left (x \right )^{2}}{2}+\textit {\_C1}}{y \left (x \right ) \left (y \left (x \right )^{3}-5\right )^{2}} = 0\right ]$ Mathematica raw input

DSolve[5*y[x] - y[x]^4 + (5*x - y[x] - 7*x*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 + 2*#1^5 + 10*x*#1^7 &
 , 1]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 + 2*#1^5 + 10*x
*#1^7 & , 2]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 + 2*#1^5
 + 10*x*#1^7 & , 3]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x*#1^4 +
 2*#1^5 + 10*x*#1^7 & , 4]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2 - 100*x
*#1^4 + 2*#1^5 + 10*x*#1^7 & , 5]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 25*#1^2
- 100*x*#1^4 + 2*#1^5 + 10*x*#1^7 & , 6]}, {y[x] -> Root[-10*C[1] + 250*x*#1 - 2
5*#1^2 - 100*x*#1^4 + 2*#1^5 + 10*x*#1^7 & , 7]}}

Maple raw input

dsolve((5*x-y(x)-7*x*y(x)^3)*diff(y(x),x)+5*y(x)-y(x)^4 = 0, y(x))

Maple raw output

[x-(-1/5*y(x)^5+5/2*y(x)^2+_C1)/y(x)/(y(x)^3-5)^2 = 0]