#### 2.511   ODE No. 511

$\left (a^2 \sqrt {x^2+y(x)^2}-x^2\right ) y'(x)^2+a^2 \sqrt {x^2+y(x)^2}+2 x y(x) y'(x)-y(x)^2=0$ Mathematica : cpu = 1.32095 (sec), leaf count = 229

$\left \{\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}+\tan ^{-1}\left (\frac {x}{y(x)}\right )=c_1,y(x)\right ]\right \}$ Maple : cpu = 8.44 (sec), leaf count = 199

$\left \{ \arctan \left ( {\frac {x}{y \left ( x \right ) }} \right ) -2\,{\frac {\sqrt {{a}^{2} \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) \left ( -{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}} \right ) }}{a\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}\sqrt {-{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}\arctan \left ( {\frac {\sqrt {-{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}{a}} \right ) }-{\it \_C1}=0,\arctan \left ( {\frac {x}{y \left ( x \right ) }} \right ) +2\,{\frac {\sqrt {{a}^{2} \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) \left ( -{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}} \right ) }}{a\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}\sqrt {-{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}\arctan \left ( {\frac {\sqrt {-{a}^{2}+\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}}{a}} \right ) }-{\it \_C1}=0 \right \}$