##### 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 = 0.90894 (sec), leaf count = 47

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

Maple
cpu = 0.193 (sec), leaf count = 217

$\left [\frac {\left (-2 y \left (x \right )+2 \sqrt {2 x +y \left (x \right )^{2}}\right ) \textit {\_C1}}{\sqrt {2 y \left (x \right )^{2}+2 x -2 y \left (x \right ) \sqrt {2 x +y \left (x \right )^{2}}+1}}+x -\frac {\left (-y \left (x \right )+\sqrt {2 x +y \left (x \right )^{2}}\right ) \arcsinh \left (-y \left (x \right )+\sqrt {2 x +y \left (x \right )^{2}}\right )}{2 \sqrt {2 y \left (x \right )^{2}+2 x -2 y \left (x \right ) \sqrt {2 x +y \left (x \right )^{2}}+1}} = 0, \frac {\left (2 y \left (x \right )+2 \sqrt {2 x +y \left (x \right )^{2}}\right ) \textit {\_C1}}{\sqrt {2 y \left (x \right )^{2}+2 x +2 y \left (x \right ) \sqrt {2 x +y \left (x \right )^{2}}+1}}+x -\frac {\left (y \left (x \right )+\sqrt {2 x +y \left (x \right )^{2}}\right ) \arcsinh \left (y \left (x \right )+\sqrt {2 x +y \left (x \right )^{2}}\right )}{2 \sqrt {2 y \left (x \right )^{2}+2 x +2 y \left (x \right ) \sqrt {2 x +y \left (x \right )^{2}}+1}} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[(-2*y(x)+2*(2*x+y(x)^2)^(1/2))/(2*y(x)^2+2*x-2*y(x)*(2*x+y(x)^2)^(1/2)+1)^(1/2)
*_C1+x-1/2*(-y(x)+(2*x+y(x)^2)^(1/2))*arcsinh(-y(x)+(2*x+y(x)^2)^(1/2))/(2*y(x)^
2+2*x-2*y(x)*(2*x+y(x)^2)^(1/2)+1)^(1/2) = 0, (2*y(x)+2*(2*x+y(x)^2)^(1/2))/(2*y
(x)^2+2*x+2*y(x)*(2*x+y(x)^2)^(1/2)+1)^(1/2)*_C1+x-1/2*(y(x)+(2*x+y(x)^2)^(1/2))
*arcsinh(y(x)+(2*x+y(x)^2)^(1/2))/(2*y(x)^2+2*x+2*y(x)*(2*x+y(x)^2)^(1/2)+1)^(1/
2) = 0]