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

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 = 13.296 (sec), leaf count = 34

\[\text {Solve}\left [\left \{x=\frac {c_1-2 \text {K$\$$1206}+2 \log (\text {K$\$$1206})}{(\text {K$\$$1206}-1)^2},\text {K$\$$1206} (\text {K$\$$1206} x+2)=y(x)\right \},\{y(x),\text {K$\$$1206}\}\right ]\]

Maple
cpu = 0.017 (sec), leaf count = 42

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac {-2\,{\it \_T}+2\,\ln \left ({\it \_T} \right ) +{\it \_C1}}{ \left ({\it \_T}-1 \right ) ^{2}}},y \left ({\it \_T} \right ) ={\frac {{\it \_T}\, \left (2\,{\it \_T}\,\ln \left ({\it \_T} \right ) +2+ \left ({\it \_C1}-4 \right ) {\it \_T} \right ) }{ \left ({\it \_T}-1 \right ) ^{2}}}] \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[{x == (-2*K$1206 + C[1] + 2*Log[K$1206])/(-1 + K$1206)^2, K$1206*(2 + K$12
06*x) == y[x]}, {y[x], K$1206}]

Maple raw input

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

Maple raw output

[x(_T) = 1/(_T-1)^2*(-2*_T+2*ln(_T)+_C1), y(_T) = _T*(2*_T*ln(_T)+2+(_C1-4)*_T)/
(_T-1)^2]