#### 2.408   ODE No. 408

$x y'(x)^2-2 y(x)+x=0$ Mathematica : cpu = 0.148725 (sec), leaf count = 97

$\left \{\text {Solve}\left [-2 \left (\frac {1}{1-\sqrt {\frac {2 y(x)}{x}-1}}+\log \left (1-\sqrt {\frac {2 y(x)}{x}-1}\right )\right )=\log (x)+c_1,y(x)\right ],\text {Solve}\left [2 \left (\frac {1}{\sqrt {\frac {2 y(x)}{x}-1}+1}+\log \left (\sqrt {\frac {2 y(x)}{x}-1}+1\right )\right )=-\log (x)+c_1,y(x)\right ]\right \}$ Maple : cpu = 0.117 (sec), leaf count = 73

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