ODE
\[ 2 y'(x)^2+x y'(x)-2 y(x)=0 \] 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 = 0.87265 (sec), leaf count = 96
\[\left \{\text {Solve}\left [\frac {x \left (\sqrt {x^2+16 y(x)}-x\right )}{16 y(x)}+\log \left (\sqrt {x^2+16 y(x)}+x\right )=c_1,y(x)\right ],\text {Solve}\left [\log (y(x))=\frac {x \left (\sqrt {x^2+16 y(x)}+x\right )}{16 y(x)}+\log \left (\sqrt {x^2+16 y(x)}+x\right )+c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.446 (sec), leaf count = 36
\[\left [y \left (x \right ) = {\mathrm e}^{2 \LambertW \left (\frac {x \,{\mathrm e}^{\frac {\textit {\_C1}}{4}}}{4}\right )-\frac {\textit {\_C1}}{2}}+\frac {x \,{\mathrm e}^{\LambertW \left (\frac {x \,{\mathrm e}^{\frac {\textit {\_C1}}{4}}}{4}\right )-\frac {\textit {\_C1}}{4}}}{2}\right ]\] Mathematica raw input
DSolve[-2*y[x] + x*y'[x] + 2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[Log[x + Sqrt[x^2 + 16*y[x]]] + (x*(-x + Sqrt[x^2 + 16*y[x]]))/(16*y[x]) == C[1], y[x]], Solve[Log[y[x]] == C[1] + Log[x + Sqrt[x^2 + 16*y[x]]] + (x*(x + Sqrt[x^2 + 16*y[x]]))/(16*y[x]), y[x]]}
Maple raw input
dsolve(2*diff(y(x),x)^2+x*diff(y(x),x)-2*y(x) = 0, y(x))
Maple raw output
[y(x) = exp(2*LambertW(1/4*x*exp(1/4*_C1))-1/2*_C1)+1/2*x*exp(LambertW(1/4*x*exp(1/4*_C1))-1/4*_C1)]