#### 2.453   ODE No. 453

$\left (a^2-1\right ) x^2 y'(x)^2+a^2 x^2+2 x y(x) y'(x)-y(x)^2=0$ Mathematica : cpu = 0.582589 (sec), leaf count = 327

$\left \{\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \tanh ^{-1}\left (\frac {-a^2-\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-a \tanh ^{-1}\left (\frac {-a^2+\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ],\text {Solve}\left [\frac {-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-a \tanh ^{-1}\left (\frac {-a^2-\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \tanh ^{-1}\left (\frac {-a^2+\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ]\right \}$ Maple : cpu = 2.691 (sec), leaf count = 229

$\left \{ {\frac {1}{2\,a} \left ( -2\,{\it \_C1}\,a+2\,a\ln \left ( x \right ) +\ln \left ( {\frac { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}{{x}^{2}}} \right ) a-2\,\sqrt {-{a}^{2}}\arctan \left ( {\frac {{a}^{2}y \left ( x \right ) }{\sqrt {-{a}^{2}}x}{\frac {1}{\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}+ \left ( -{a}^{2}+1 \right ) {x}^{2}}{{x}^{2}}}}}}} \right ) +2\,\ln \left ( {\frac {1}{x} \left ( \sqrt {{\frac {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}x+y \left ( x \right ) \right ) } \right ) \right ) }=0,{\frac {1}{2\,a} \left ( -2\,{\it \_C1}\,a+2\,a\ln \left ( x \right ) +\ln \left ( {\frac { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}{{x}^{2}}} \right ) a+2\,\sqrt {-{a}^{2}}\arctan \left ( {\frac {{a}^{2}y \left ( x \right ) }{\sqrt {-{a}^{2}}x}{\frac {1}{\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}+ \left ( -{a}^{2}+1 \right ) {x}^{2}}{{x}^{2}}}}}}} \right ) -2\,\ln \left ( {\frac {1}{x} \left ( \sqrt {{\frac {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}x+y \left ( x \right ) \right ) } \right ) \right ) }=0 \right \}$