4.18.2 $x{y}^{\prime }(x{)}^2+2y^{\prime }(x)-y(x)=0$

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

[_rational, _dAlembert]

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

Mathematica ✓
cpu = 13.296 (sec), leaf count = 34

$$\text{Solve}[\{x=\frac{c_1-2\text{K}\mathrm{\$}\text{1206}+2\log (\text{K}\mathrm{\$}\text{1206})}{(\text{K}\mathrm{\$}\text{1206}-1{)}^2},\text{K}\mathrm{\$}\text{1206}(\text{K}\mathrm{\$}\text{1206}x+2)=y(x)\},\{y(x),\text{K}\mathrm{\$}\text{1206}\}]$$

Maple ✓
cpu = 0.017 (sec), leaf count = 42

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\frac{-2\mathit{\_}\mathit{T}+2\ln \left(\mathit{\_}\mathit{T}\right)+\mathit{\_}\mathit{C}\mathit{1}}{{(\mathit{\_}\mathit{T}-1)}^2},y\left(\mathit{\_}\mathit{T}\right)=\frac{\mathit{\_}\mathit{T}(2\mathit{\_}\mathit{T}\ln \left(\mathit{\_}\mathit{T}\right)+2+(\mathit{\_}\mathit{C}\mathit{1}-4)\mathit{\_}\mathit{T})}{{(\mathit{\_}\mathit{T}-1)}^2}]\}$$ Mathematica raw input

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

Mathematica raw output

Solve[{x == (-2*K$1206 + C[1] + 2*Log[K$1206])/(-1 + K$1206)^2, K$1206*(2 + K$12
06*x) == y[x]}, {y[x], K$1206}]

Maple raw input

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

Maple raw output

[x(_T) = 1/(_T-1)^2*(-2*_T+2*ln(_T)+_C1), y(_T) = _T*(2*_T*ln(_T)+2+(_C1-4)*_T)/
(_T-1)^2]