4.11.31 \(x (y(x)+x) y'(x)=x^2+y(x)^2\)

ODE
\[ x (y(x)+x) y'(x)=x^2+y(x)^2 \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0218283 (sec), leaf count = 30

\[\left \{\left \{y(x)\to 2 x W\left (\frac {e^{\frac {1}{2} \left (c_1-1\right )}}{2 \sqrt {x}}\right )+x\right \}\right \}\]

Maple
cpu = 0.011 (sec), leaf count = 31

\[ \left \{ -2\,\ln \left ({\frac {y \relax (x ) -x}{x}} \right ) -{\frac {y \relax (x ) }{x}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-2*ln((y(x)-x)/x)-y(x)/x-ln(x)-_C1 = 0