#### 2.517   ODE No. 517

$\left (x^2+y(x)^2\right ) f\left (\frac {y(x)}{\sqrt {x^2+y(x)^2}}\right ) \left (y'(x)^2+1\right )-\left (x y'(x)-y(x)\right )^2=0$ Mathematica : cpu = 0.457567 (sec), leaf count = 283

$\left \{\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right ) K[1]^2+f\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )-1}{\sqrt {f\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )} (K[1]-i) (K[1]+i) \left (\sqrt {f\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )} K[1]+i \sqrt {f\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )-1}\right )}dK[1]=-\log (x)+c_1,y(x)\right ],\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {K[2]}{\sqrt {K[2]^2+1}}\right ) K[2]^2+f\left (\frac {K[2]}{\sqrt {K[2]^2+1}}\right )-1}{\sqrt {f\left (\frac {K[2]}{\sqrt {K[2]^2+1}}\right )} (K[2]-i) (K[2]+i) \left (\sqrt {f\left (\frac {K[2]}{\sqrt {K[2]^2+1}}\right )} K[2]-i \sqrt {f\left (\frac {K[2]}{\sqrt {K[2]^2+1}}\right )-1}\right )}dK[2]=-\log (x)+c_1,y(x)\right ]\right \}$ Maple : cpu = 3.031 (sec), leaf count = 157

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