4.24.7 $a(\log (y^{\prime }(x))-y^{\prime }(x))+y(x)-x=0$

ODE
$$a(\log (y^{\prime }(x))-y^{\prime }(x))+y(x)-x=0$$ ODE Classification

[[_homogeneous, `class C`], _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica ✓
cpu = 0.0544786 (sec), leaf count = 22

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

Maple ✓
cpu = 0.008 (sec), leaf count = 25

$$\{y\left(x\right)=a+x,[x\left(\mathit{\_}\mathit{T}\right)=a\ln \left(\mathit{\_}\mathit{T}\right)+\mathit{\_}\mathit{C}\mathit{1},y\left(\mathit{\_}\mathit{T}\right)=a\mathit{\_}\mathit{T}+\mathit{\_}\mathit{C}\mathit{1}]\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> a*E^((x - C[1])/a) + C[1]}}

Maple raw input

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

Maple raw output

y(x) = a+x, [x(_T) = a*ln(_T)+_C1, y(_T) = _T*a+_C1]