#### 2.512   ODE No. 512

$\left (a \left (x^2+y(x)^2\right )^{3/2}-x^2\right ) y'(x)^2+a \left (x^2+y(x)^2\right )^{3/2}+2 x y(x) y'(x)-y(x)^2=0$ Mathematica : cpu = 5.92611 (sec), leaf count = 713

$\left \{\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )-\frac {i \sqrt {a \left (\left (x^2+y(x)^2\right )^{5/2}-a \left (x^2+y(x)^2\right )^3\right )} \left (\sqrt {2} \left (\log \left (\frac {a^{3/2} \left (3 i \sqrt {2} a \sqrt {x^2+y(x)^2}+4 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}-i \sqrt {2}\right )}{4 a \sqrt {x^2+y(x)^2}+4}\right )-\log \left (\frac {-3 i \sqrt {2} a^{3/2} \sqrt {x^2+y(x)^2}-4 a \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i \sqrt {2} \sqrt {a}}{4 a \sqrt {x^2+y(x)^2}+4}\right )\right )+2 \log \left (\frac {-2 i a \sqrt {x^2+y(x)^2}+2 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i}{\sqrt {a}}\right )\right )}{2 \sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}}=c_1,y(x)\right ],\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )+\frac {i \sqrt {a \left (\left (x^2+y(x)^2\right )^{5/2}-a \left (x^2+y(x)^2\right )^3\right )} \left (\sqrt {2} \left (\log \left (\frac {a^{3/2} \left (3 i \sqrt {2} a \sqrt {x^2+y(x)^2}+4 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}-i \sqrt {2}\right )}{4 a \sqrt {x^2+y(x)^2}+4}\right )-\log \left (\frac {-3 i \sqrt {2} a^{3/2} \sqrt {x^2+y(x)^2}-4 a \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i \sqrt {2} \sqrt {a}}{4 a \sqrt {x^2+y(x)^2}+4}\right )\right )+2 \log \left (\frac {-2 i a \sqrt {x^2+y(x)^2}+2 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i}{\sqrt {a}}\right )\right )}{2 \sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}}=c_1,y(x)\right ]\right \}$ Maple : cpu = 4.408 (sec), leaf count = 137

$\left \{ y \left ( x \right ) ={x \left ( \tan \left ( {\it RootOf} \left ( -{\it \_Z}+\int ^{{\frac {{x}^{2} \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) }{ \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}}}}\!-{\frac {1}{2\,{{\it \_a}}^{2} \left ( {\it \_a}\,{a}^{2}-1 \right ) } \left ( a\sqrt {{\it \_a}}+1 \right ) \sqrt {-{{\it \_a}}^{3}{a}^{2}+{{\it \_a}}^{{\frac {5}{2}}}a}}{d{\it \_a}}+{\it \_C1} \right ) \right ) \right ) ^{-1}},y \left ( x \right ) ={x \left ( \tan \left ( {\it RootOf} \left ( -{\it \_Z}+\int ^{{\frac {{x}^{2} \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) }{ \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}}}}\!{\frac {1}{2\,{{\it \_a}}^{2} \left ( {\it \_a}\,{a}^{2}-1 \right ) } \left ( a\sqrt {{\it \_a}}+1 \right ) \sqrt {-{{\it \_a}}^{3}{a}^{2}+{{\it \_a}}^{{\frac {5}{2}}}a}}{d{\it \_a}}+{\it \_C1} \right ) \right ) \right ) ^{-1}} \right \}$