#### 2.558   ODE No. 558

$a x \sqrt {y'(x)^2+1}+x y'(x)-y(x)=0$ Mathematica : cpu = 0.561397 (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 = 0.112 (sec), leaf count = 223

$\left \{ x-{{\it \_C1}{{\rm e}^{{\frac {1}{a}{\it Arcsinh} \left ( {\frac {1}{ \left ( {a}^{2}-1 \right ) x} \left ( \sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}a+y \left ( x \right ) \right ) } \right ) }}}{\frac {1}{\sqrt {{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2}{x}^{2}} \left ( -{a}^{2}{x}^{2}+{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\,\sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}ay \left ( x \right ) +{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0,x-{{\it \_C1}{{\rm e}^{-{\frac {1}{a}{\it Arcsinh} \left ( {\frac {1}{ \left ( {a}^{2}-1 \right ) x} \left ( \sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}a-y \left ( x \right ) \right ) } \right ) }}}{\frac {1}{\sqrt {-{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2}{x}^{2}} \left ( {a}^{2}{x}^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\,\sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}ay \left ( x \right ) -{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0 \right \}$