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

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0135442 (sec), leaf count = 18

\[\left \{\left \{y(x)\to -x \log \left (-c_1-\log (x)\right )\right \}\right \}\]

Maple
cpu = 0.01 (sec), leaf count = 18

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

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

Mathematica raw output

{{y[x] -> -(x*Log[-C[1] - Log[x]])}}

Maple raw input

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

Maple raw output

-_C1+1/exp(y(x)/x)+ln(x) = 0