#### 2.397   ODE No. 397

$-2 x^3 y(x)^2 y'(x)-4 x^2 y(x)^3+y'(x)^2=0$ Mathematica : cpu = 0.411392 (sec), leaf count = 143

$\left \{\text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+4} y(x)^{3/2} \sinh ^{-1}\left (\frac {1}{2} x^2 \sqrt {y(x)}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ],\text {Solve}\left [\frac {x y(x)^{3/2} \sqrt {x^4 y(x)+4} \sinh ^{-1}\left (\frac {1}{2} x^2 \sqrt {y(x)}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ]\right \}$ Maple : cpu = 0.802 (sec), leaf count = 128

$\left \{ y \left ( x \right ) ={\frac {-2\,\sqrt {2}{x}^{2}-2\,{\it \_C1}}{2\,{\it \_C1}\,{x}^{4}-{{\it \_C1}}^{3}}},y \left ( x \right ) ={\frac {2\,\sqrt {2}{x}^{2}-2\,{\it \_C1}}{2\,{\it \_C1}\,{x}^{4}-{{\it \_C1}}^{3}}},y \left ( x \right ) ={\frac { \left ( \sqrt {2}{x}^{2}{\it \_C1}-2 \right ) {{\it \_C1}}^{2}}{2\,{{\it \_C1}}^{2}{x}^{4}-4}},y \left ( x \right ) =-4\,{x}^{-4},y \left ( x \right ) =-{\frac { \left ( \sqrt {2}{x}^{2}{\it \_C1}+2 \right ) {{\it \_C1}}^{2}}{2\,{{\it \_C1}}^{2}{x}^{4}-4}} \right \}$