4.18.1 $x{y}^{\prime }(x{)}^2+y^{\prime }(x)=y(x)$

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

[_rational, _dAlembert]

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

Mathematica ✓
cpu = 1.23018 (sec), leaf count = 32

$$\text{Solve}[\{x=\frac{c_1-\text{K}\mathrm{\$}\text{1057}+\log (\text{K}\mathrm{\$}\text{1057})}{(\text{K}\mathrm{\$}\text{1057}-1{)}^2},{\text{K}\mathrm{\$}\text{1057}}^{2}x+\text{K}\mathrm{\$}\text{1057}=y(x)\},\{y(x),\text{K}\mathrm{\$}\text{1057}\}]$$

Maple ✓
cpu = 0.018 (sec), leaf count = 39

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

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

Mathematica raw output

Solve[{x == (-K$1057 + C[1] + Log[K$1057])/(-1 + K$1057)^2, K$1057 + K$1057^2*x 
== y[x]}, {y[x], K$1057}]

Maple raw input

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

Maple raw output

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