#### 2.388   ODE No. 388

$y'(x)^2-2 y(x) y'(x)-2 x=0$ Mathematica : cpu = 0.493027 (sec), leaf count = 53

$\text {Solve}\left [\left \{x=\frac {c_1 \text {K\305940}}{\sqrt {\text {K\305940}^2+1}}+\frac {\text {K\305940} \sinh ^{-1}(\text {K\305940})}{2 \sqrt {\text {K\305940}^2+1}},y(x)=\frac {\text {K\305940}}{2}-\frac {x}{\text {K\305940}}\right \},\{y(x),\text {K\305940}\}\right ]$ Maple : cpu = 0.162 (sec), leaf count = 223

$\left \{ {1 \left ( \left ( -{\frac {y \left ( x \right ) }{2}}-{\frac {1}{2}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x}} \right ) {\it Arcsinh} \left ( y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x} \right ) +x\sqrt {2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,x+2\,y \left ( x \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x}+1}+2\,{\it \_C1}\,y \left ( x \right ) +2\,{\it \_C1}\,\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x} \right ) {\frac {1}{\sqrt {2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,x+2\,y \left ( x \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x}+1}}}}=0,{1 \left ( \left ( {\frac {y \left ( x \right ) }{2}}-{\frac {1}{2}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x}} \right ) {\it Arcsinh} \left ( -y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x} \right ) +x\sqrt {2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,x-2\,y \left ( x \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x}+1}-2\,{\it \_C1}\,y \left ( x \right ) +2\,{\it \_C1}\,\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x} \right ) {\frac {1}{\sqrt {2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,x-2\,y \left ( x \right ) \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+2\,x}+1}}}}=0 \right \}$