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 = 0.0353911 (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 = 0.016 (sec), leaf count = 32

\[ \left \{ -\ln \left ({\frac {y \relax (x ) }{x}} \right ) +{\frac {{x}^{2}}{2\,a \left (y \relax (x ) \right ) ^{2}}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-ln(y(x)/x)+1/2/a*x^2/y(x)^2-ln(x)-_C1 = 0