4.9.39 \((y(x)+x) y'(x)-y(x)+x=0\)

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0301354 (sec), leaf count = 31

\[\text {Solve}\left [\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )+\tan ^{-1}\left (\frac {y(x)}{x}\right )+\log (x)=c_1,y(x)\right ]\]

Maple
cpu = 0.013 (sec), leaf count = 35

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

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

Mathematica raw output

Solve[ArcTan[y[x]/x] + Log[x] + Log[1 + y[x]^2/x^2]/2 == C[1], y[x]]

Maple raw input

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

Maple raw output

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