##### 4.13.36 $$\left (a y(x)^2+x^2\right ) y'(x)=x y(x)$$

ODE
$\left (a y(x)^2+x^2\right ) y'(x)=x y(x)$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.374014 (sec), leaf count = 66

$\left \{\left \{y(x)\to -\frac {x}{\sqrt {a} \sqrt {W\left (\frac {x^2 e^{-\frac {2 c_1}{a}}}{a}\right )}}\right \},\left \{y(x)\to \frac {x}{\sqrt {a} \sqrt {W\left (\frac {x^2 e^{-\frac {2 c_1}{a}}}{a}\right )}}\right \}\right \}$

Maple
cpu = 0.156 (sec), leaf count = 23

$\left [y \left (x \right ) = \sqrt {\frac {1}{a \LambertW \left (\frac {\textit {\_C1} \,x^{2}}{a}\right )}}\, x\right ]$ Mathematica raw input

DSolve[(x^2 + a*y[x]^2)*y'[x] == x*y[x],y[x],x]

Mathematica raw output

{{y[x] -> -(x/(Sqrt[a]*Sqrt[ProductLog[x^2/(a*E^((2*C[1])/a))]]))}, {y[x] -> x/(
Sqrt[a]*Sqrt[ProductLog[x^2/(a*E^((2*C[1])/a))]])}}

Maple raw input

dsolve((x^2+a*y(x)^2)*diff(y(x),x) = x*y(x), y(x))

Maple raw output

[y(x) = (1/a/LambertW(1/a*_C1*x^2))^(1/2)*x]