4.18.3 $x{y}^{\prime }(x{)}^2-2y^{\prime }(x)-y(x)=0$

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 = 30.7129 (sec), leaf count = 39

$$\text{Solve}[\{\text{K}\mathrm{\$}\text{1374}x=\frac{y(\text{K}\mathrm{\$}\text{1374})}{\text{K}\mathrm{\$}\text{1374}}+2,y(x)=\frac{\text{K}\mathrm{\$}\text{1374}(c_1\text{K}\mathrm{\$}\text{1374}-2\text{K}\mathrm{\$}\text{1374}\log (\text{K}\mathrm{\$}\text{1374})-2)}{(\text{K}\mathrm{\$}\text{1374}-1{)}^2}\},\{y(x),\text{K}\mathrm{\$}\text{1374}\}]$$

Maple ✓
cpu = 0.016 (sec), leaf count = 42

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

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

Mathematica raw output

Solve[{K$1374*x == 2 + y[K$1374]/K$1374, y[x] == (K$1374*(-2 + K$1374*C[1] - 2*K
$1374*Log[K$1374]))/(-1 + K$1374)^2}, {y[x], K$1374}]

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