4.9.37 \((y(x)+x) y'(x)+y(x)=0\)

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0217192 (sec), leaf count = 47

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

Maple
cpu = 0.033 (sec), leaf count = 16

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

x+1/2*y(x)-1/y(x)*_C1 = 0