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

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

[_dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(x\)

Mathematica
cpu = 0.718113 (sec), leaf count = 41

\[\text {Solve}\left [\left \{x=\frac {\text {K$\$$219290} \left (2 c_1+\sinh ^{-1}(\text {K$\$$219290})\right )}{2 \sqrt {\text {K$\$$219290}^2+1}},\text {K$\$$219290}=2 \left (\frac {x}{\text {K$\$$219290}}+y(x)\right )\right \},\{y(x),\text {K$\$$219290}\}\right ]\]

Maple
cpu = 0.029 (sec), leaf count = 42

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

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

Mathematica raw output

Solve[{x == (K$219290*(ArcSinh[K$219290] + 2*C[1]))/(2*Sqrt[1 + K$219290^2]), K$
219290 == 2*(x/K$219290 + y[x])}, {y[x], K$219290}]

Maple raw input

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

Maple raw output

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