4.5.17 \(x y'(x)=x e^{\frac {y(x)}{x}}+y(x)+x\)

ODE
\[ x y'(x)=x e^{\frac {y(x)}{x}}+y(x)+x \] ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0242288 (sec), leaf count = 21

\[\left \{\left \{y(x)\to -x \log \left (\frac {e^{-c_1}}{x}-1\right )\right \}\right \}\]

Maple
cpu = 0.022 (sec), leaf count = 31

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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