4.18.5 $x{y}^{\prime }(x{)}^2+x{y}^{\prime }(x)-y(x)=0$

ODE
$$x{y}^{\prime }(x{)}^2+x{y}^{\prime }(x)-y(x)=0$$ ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for $y$

Mathematica ✓
cpu = 1.03203 (sec), leaf count = 180

$$\{\text{Solve}[\frac{x(\sqrt{\frac{4y(x)}{x}+1}-1)((\sqrt{\frac{4y(x)}{x}+1}-1)\log (\sqrt{\frac{4y(x)}{x}+1}-1)-1)}{2x(\sqrt{\frac{4y(x)}{x}+1}-1)-4y(x)}=c_1+\frac{\log (x)}{2},y(x)],\text{Solve}[\frac{x(\sqrt{\frac{4y(x)}{x}+1}+1)((\sqrt{\frac{4y(x)}{x}+1}+1)\log (\sqrt{\frac{4y(x)}{x}+1}+1)+1)}{2(x\sqrt{\frac{4y(x)}{x}+1}+2y(x)+x)}+\frac{\log (x)}{2}=c_1,y(x)]\}$$

Maple ✓
cpu = 0.021 (sec), leaf count = 29

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\frac{{\mathrm{e}}^{{\mathit{\_}\mathit{T}}^{-1}}\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{T}}^2},y\left(\mathit{\_}\mathit{T}\right)=\frac{(\mathit{\_}\mathit{T}+1){\mathrm{e}}^{{\mathit{\_}\mathit{T}}^{-1}}\mathit{\_}\mathit{C}\mathit{1}}{\mathit{\_}\mathit{T}}]\}$$ Mathematica raw input

DSolve[-y[x] + x*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[(x*(-1 + Sqrt[1 + (4*y[x])/x])*(-1 + Log[-1 + Sqrt[1 + (4*y[x])/x]]*(-1 +
 Sqrt[1 + (4*y[x])/x])))/(-4*y[x] + 2*x*(-1 + Sqrt[1 + (4*y[x])/x])) == C[1] + L
og[x]/2, y[x]], Solve[Log[x]/2 + (x*(1 + Sqrt[1 + (4*y[x])/x])*(1 + Log[1 + Sqrt
[1 + (4*y[x])/x]]*(1 + Sqrt[1 + (4*y[x])/x])))/(2*(x + 2*y[x] + x*Sqrt[1 + (4*y[
x])/x])) == C[1], y[x]]}

Maple raw input

dsolve(x*diff(y(x),x)^2+x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = exp(1/_T)/_T^2*_C1, y(_T) = 1/_T*(_T+1)*exp(1/_T)*_C1]