4.11.29 \(x (y(x)+x) y'(x)+y(x)^2=0\)

ODE
\[ x (y(x)+x) y'(x)+y(x)^2=0 \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0367497 (sec), leaf count = 75

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

Maple
cpu = 0.021 (sec), leaf count = 33

\[ \left \{ -\ln \left ({\frac {y \relax (x ) }{x}} \right ) +{\frac {1}{2}\ln \left ({\frac {x+2\,y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]^2 + x*(x + y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(x+y(x))*diff(y(x),x)+y(x)^2 = 0, y(x),'implicit')

Maple raw output

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