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

ODE
$$x{y}^{\prime }(x)=x{e}^{\frac{y(x)}{x}}+y(x)+x$$ ODE Classification

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

Book solution method
Homogeneous equation

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

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

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

$$\{-\ln ({\mathrm{e}}^{\frac{y\left(x\right)}{x}}+1)+\ln \left({\mathrm{e}}^{\frac{y\left(x\right)}{x}}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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