##### 4.14.4 $$2 x \left (5 x^2+y(x)^2\right ) y'(x)=x^2 y(x)-y(x)^3$$

ODE
$2 x \left (5 x^2+y(x)^2\right ) y'(x)=x^2 y(x)-y(x)^3$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.375279 (sec), leaf count = 216

$\left \{\left \{y(x)\to \text {Root}\left [-\text {\#1}^5+\frac {\text {\#1}^2 e^{3 c_1}}{x^{3/2}}+3 e^{3 c_1} \sqrt {x}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {\#1}^5+\frac {\text {\#1}^2 e^{3 c_1}}{x^{3/2}}+3 e^{3 c_1} \sqrt {x}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {\#1}^5+\frac {\text {\#1}^2 e^{3 c_1}}{x^{3/2}}+3 e^{3 c_1} \sqrt {x}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {\#1}^5+\frac {\text {\#1}^2 e^{3 c_1}}{x^{3/2}}+3 e^{3 c_1} \sqrt {x}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [-\text {\#1}^5+\frac {\text {\#1}^2 e^{3 c_1}}{x^{3/2}}+3 e^{3 c_1} \sqrt {x}\& ,5\right ]\right \}\right \}$

Maple
cpu = 0.336 (sec), leaf count = 29

$\left [y \left (x \right ) = \RootOf \left (x^{9} \textit {\_C1} \,\textit {\_Z}^{45}-\textit {\_Z}^{18}-6 \textit {\_Z}^{9}-9\right )^{\frac {9}{2}} x\right ]$ Mathematica raw input

DSolve[2*x*(5*x^2 + y[x]^2)*y'[x] == x^2*y[x] - y[x]^3,y[x],x]

Mathematica raw output

{{y[x] -> Root[3*E^(3*C[1])*Sqrt[x] + (E^(3*C[1])*#1^2)/x^(3/2) - #1^5 & , 1]},
{y[x] -> Root[3*E^(3*C[1])*Sqrt[x] + (E^(3*C[1])*#1^2)/x^(3/2) - #1^5 & , 2]}, {
y[x] -> Root[3*E^(3*C[1])*Sqrt[x] + (E^(3*C[1])*#1^2)/x^(3/2) - #1^5 & , 3]}, {y
[x] -> Root[3*E^(3*C[1])*Sqrt[x] + (E^(3*C[1])*#1^2)/x^(3/2) - #1^5 & , 4]}, {y[
x] -> Root[3*E^(3*C[1])*Sqrt[x] + (E^(3*C[1])*#1^2)/x^(3/2) - #1^5 & , 5]}}

Maple raw input

dsolve(2*x*(5*x^2+y(x)^2)*diff(y(x),x) = x^2*y(x)-y(x)^3, y(x))

Maple raw output

[y(x) = RootOf(_C1*_Z^45*x^9-_Z^18-6*_Z^9-9)^(9/2)*x]