4.16.38 \(y'(x)^2+x y'(x)-y(x)+x=0\)

ODE
\[ y'(x)^2+x y'(x)-y(x)+x=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica ✓
cpu = 0.601162 (sec), leaf count = 122

\[\left \{\text {Solve}\left [c_1+2 \log \left (-x^2-4 y(x)+4 x+4\right )+\sqrt {4 y(x)+(x-4) x}=\log \left (x^2+4 y(x)+4 \sqrt {4 y(x)+(x-4) x}-4 x+4\right )+x,y(x)\right ],\text {Solve}\left [\sqrt {4 y(x)+(x-4) x}+x=c_1+\log \left (x^2+4 y(x)+4 \sqrt {4 y(x)+(x-4) x}-4 x+4\right ),y(x)\right ]\right \}\]

Maple ✓
cpu = 0.015 (sec), leaf count = 48

\[ \left \{ [x \left ( {\it \_T} \right ) = \left ( \left ( -2\,{\it \_T}+2 \right ) {{\rm e}^{{\it \_T}}}+{\it \_C1} \right ) {{\rm e}^{-{\it \_T}}},y \left ( {\it \_T} \right ) = \left ( -2\,{{\it \_T}}^{2}{{\rm e}^{{\it \_T}}}+{\it \_T}\,{\it \_C1}+{\it \_C1}+2\,{{\rm e}^{{\it \_T}}} \right ) {{\rm e}^{-{\it \_T}}}+{{\it \_T}}^{2}] \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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