4.9.40 \((y(x)+x) y'(x)=x-y(x)\)

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0226282 (sec), leaf count = 51

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

Maple
cpu = 0.011 (sec), leaf count = 33

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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