#### 2.515   ODE No. 515

$f\left (x^2+y(x)^2\right ) \left (y'(x)^2+1\right )-\left (x y'(x)-y(x)\right )^2=0$ Mathematica : cpu = 2.04037 (sec), leaf count = 1922

$\left \{\text {Solve}\left [\int _1^x\left (\frac {\sqrt {f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )} K[1]}{f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2\right )}-\frac {\sqrt {f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )} K[1]}{f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )}+\frac {y(x)}{K[1]^2+y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {x}{x^2+K[2]^2}-\int _1^x\left (-\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}-\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} f'\left (K[1]^2+K[2]^2\right ) K[2]}{f\left (K[1]^2+K[2]^2\right )^2 \left (K[1]^2+K[2]^2\right )}+\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} f'\left (K[1]^2+K[2]^2\right ) K[2]}{f\left (K[1]^2+K[2]^2\right )^2 \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}-\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} K[2]}{f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2\right )^2}+\frac {K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )}{f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )^2}+\frac {K[1] \left (2 K[2] \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) f'\left (K[1]^2+K[2]^2\right )+f\left (K[1]^2+K[2]^2\right ) \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )\right )}{2 f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2\right ) \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}}-\frac {K[1] \left (2 K[2] \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) f'\left (K[1]^2+K[2]^2\right )+f\left (K[1]^2+K[2]^2\right ) \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )\right )}{2 f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}}+\frac {1}{K[1]^2+K[2]^2}\right )dK[1]+\frac {K[2] \sqrt {f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}}{f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2\right )}-\frac {K[2] \sqrt {f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}}{f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}\right )dK[2]=c_1,y(x)\right ],\text {Solve}\left [\int _1^x\left (-\frac {\sqrt {f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )} K[3]}{f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2\right )}+\frac {\sqrt {f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )} K[3]}{f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )}+\frac {y(x)}{K[3]^2+y(x)^2}\right )dK[3]+\int _1^{y(x)}\left (-\frac {x}{x^2+K[4]^2}-\int _1^x\left (-\frac {2 K[4]^2}{\left (K[3]^2+K[4]^2\right )^2}+\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} f'\left (K[3]^2+K[4]^2\right ) K[4]}{f\left (K[3]^2+K[4]^2\right )^2 \left (K[3]^2+K[4]^2\right )}-\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} f'\left (K[3]^2+K[4]^2\right ) K[4]}{f\left (K[3]^2+K[4]^2\right )^2 \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}+\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} K[4]}{f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2\right )^2}-\frac {K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )}{f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )^2}-\frac {K[3] \left (2 K[4] \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) f'\left (K[3]^2+K[4]^2\right )+f\left (K[3]^2+K[4]^2\right ) \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )\right )}{2 f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2\right ) \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}}+\frac {K[3] \left (2 K[4] \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) f'\left (K[3]^2+K[4]^2\right )+f\left (K[3]^2+K[4]^2\right ) \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )\right )}{2 f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}}+\frac {1}{K[3]^2+K[4]^2}\right )dK[3]-\frac {K[4] \sqrt {f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}}{f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2\right )}+\frac {K[4] \sqrt {f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}}{f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}\right )dK[4]=c_1,y(x)\right ]\right \}$ Maple : cpu = 0.943 (sec), leaf count = 113

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