2.468   ODE No. 468

\[ -4 a^2 x y'(x)+a^2 y(x)+y(x) y'(x)^2=0 \] Mathematica : cpu = 6.7618 (sec), leaf count = 753


\[\left \{\text {Solve}\left [\frac {8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )+\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )-2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )+\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )-8 \tan ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}=-\log (x)+c_1,y(x)\right ],\text {Solve}\left [\frac {\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (-4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )+2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )+\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )+8 \tan ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )\right )-8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}=-\log (x)+c_1,y(x)\right ]\right \}\] Maple : cpu = 0.086 (sec), leaf count = 181


\[-\frac {c_{1} x}{y \relax (x ) a \left (\frac {a^{2} \left (2 a^{2} x^{2}+\sqrt {4 a^{2} x^{2}-y \relax (x )^{2}}\, a x -y \relax (x )^{2}\right )}{y \relax (x )^{2}}\right )^{\frac {1}{3}} \left (\frac {\left (2 a x +\sqrt {4 a^{2} x^{2}-y \relax (x )^{2}}\right ) a}{y \relax (x )}\right )^{\frac {1}{3}}}+x = 0\]