4.18.3 \(x y'(x)^2-2 y'(x)-y(x)=0\)

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

[_rational, _dAlembert]

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

Mathematica ✓
cpu = 2.7013 (sec), leaf count = 40

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

Maple ✓
cpu = 0.851 (sec), leaf count = 63

\[[y \left (x \right ) = x \,{\mathrm e}^{2 \RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_C1} -2 \textit {\_Z} -x \right )}-2 \,{\mathrm e}^{\RootOf \left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_C1} -2 \textit {\_Z} -x \right )}]\] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = x*exp(2*RootOf(-x*exp(2*_Z)+2*x*exp(_Z)+2*exp(_Z)+_C1-2*_Z-x))-2*exp(Roo
tOf(-x*exp(2*_Z)+2*x*exp(_Z)+2*exp(_Z)+_C1-2*_Z-x))]