#### 2.468   ODE No. 468

$-4 a^2 x y'(x)+a^2 y(x)+y(x) y'(x)^2=0$ Mathematica : cpu = 3.57039 (sec), leaf count = 609

$\left \{\text {Solve}\left [\frac {-4 \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )+2 \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )+\sqrt {2 a-\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\sqrt {2 a+\frac {y(x)}{x}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )+8 \tan ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )-8 \sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \sin ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{2 \sqrt {a}}\right )\right )}{6 \sqrt {2 a-\frac {y(x)}{x}} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}=c_1-\log (x),y(x)\right ],\text {Solve}\left [\frac {4 \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )-2 \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )+\sqrt {2 a-\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\sqrt {2 a+\frac {y(x)}{x}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )-8 \tan ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )+8 \sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \sin ^{-1}\left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{2 \sqrt {a}}\right )\right )}{6 \sqrt {2 a-\frac {y(x)}{x}} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}=c_1-\log (x),y(x)\right ]\right \}$ Maple : cpu = 0.146 (sec), leaf count = 181

$\left \{ -{\frac {{\it \_C1}\,x}{ay \left ( x \right ) }{\frac {1}{\sqrt [3]{{\frac {{a}^{2}}{ \left ( y \left ( x \right ) \right ) ^{2}} \left ( 2\,{a}^{2}{x}^{2}+\sqrt {4\,{a}^{2}{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}ax- \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}{\frac {1}{\sqrt [3]{{\frac {a}{y \left ( x \right ) } \left ( 2\,ax+\sqrt {4\,{a}^{2}{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}} \right ) }}}}}+x=0,-{\frac {{\it \_C1}\,x}{ay \left ( x \right ) }{\frac {1}{\sqrt [3]{-4\,{\frac {{a}^{2} \left ( -2\,{a}^{2}{x}^{2}+\sqrt {4\,{a}^{2}{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}ax+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }{ \left ( y \left ( x \right ) \right ) ^{2}}}}}}{\frac {1}{\sqrt [3]{-{\frac {a}{y \left ( x \right ) } \left ( -2\,ax+\sqrt {4\,{a}^{2}{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}} \right ) }}}}}+x=0 \right \}$