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 = 0.468759 (sec), leaf count = 102
\[\left \{\text {Solve}\left [\frac {x \left (\sqrt {x^2+16 y(x)}-x\right )}{16 y(x)}+\log \left (\sqrt {x^2+16 y(x)}+x\right )=c_1,y(x)\right ],\text {Solve}\left [\frac {16 y(x) \left (c_1+\log \left (\sqrt {x^2+16 y(x)}+x\right )-\log (y(x))\right )+x \left (\sqrt {x^2+16 y(x)}+x\right )}{y(x)}=0,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.015 (sec), leaf count = 28
\[ \left \{ [x \left ( {\it \_T} \right ) ={\it \_T}\, \left ( 4\,\ln \left ( {\it \_T} \right ) +{\it \_C1} \right ) ,y \left ( {\it \_T} \right ) ={\frac {{{\it \_T}}^{2} \left ( 2+4\,\ln \left ( {\it \_T} \right ) +{\it \_C1} \right ) }{2}}] \right \} \] 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[(16*(C[1] - Log[y[x]] + Log[x + Sqrt[x^2 + 16*y[x]]])*y[x] + x*(x + Sqrt[x^2 + 16*y[x]]))/y[x] == 0, y[x]]}
Maple raw input
dsolve(2*diff(y(x),x)^2+x*diff(y(x),x)-2*y(x) = 0, y(x),'implicit')
Maple raw output
[x(_T) = _T*(4*ln(_T)+_C1), y(_T) = 1/2*_T^2*(2+4*ln(_T)+_C1)]