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 = 30.7129 (sec), leaf count = 39

\[\text {Solve}\left [\left \{\text {K$\$$1374} x=\frac {y(\text {K$\$$1374})}{\text {K$\$$1374}}+2,y(x)=\frac {\text {K$\$$1374} \left (c_1 \text {K$\$$1374}-2 \text {K$\$$1374} \log (\text {K$\$$1374})-2\right )}{(\text {K$\$$1374}-1)^2}\right \},\{y(x),\text {K$\$$1374}\}\right ]\]

Maple
cpu = 0.016 (sec), leaf count = 42

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

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

Mathematica raw output

Solve[{K$1374*x == 2 + y[K$1374]/K$1374, y[x] == (K$1374*(-2 + K$1374*C[1] - 2*K
$1374*Log[K$1374]))/(-1 + K$1374)^2}, {y[x], K$1374}]

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