#### 2.416   ODE No. 416

$x y'(x)^2+(y(x)-3 x) y'(x)+y(x)=0$ Mathematica : cpu = 1.27176 (sec), leaf count = 383

$\left \{\text {Solve}\left [\frac {1}{8} \left (-\sqrt {\frac {\frac {y(x)}{x}-9}{\frac {y(x)}{x}-1}} \left (\frac {y(x)}{x}-1\right )+\sqrt {\frac {y(x)}{x}-9} \sqrt {\frac {y(x)}{x}-1}-3 \log \left (\frac {y(x)}{x}\right )-\frac {10 \sqrt {\frac {y(x)}{x}-9} \sin ^{-1}\left (\frac {\sqrt {9-\frac {y(x)}{x}}}{2 \sqrt {2}}\right )}{\sqrt {9-\frac {y(x)}{x}}}+6 \tanh ^{-1}\left (\frac {1}{3} \sqrt {\frac {\frac {y(x)}{x}-9}{\frac {y(x)}{x}-1}}\right )+8 \tanh ^{-1}\left (\sqrt {\frac {\frac {y(x)}{x}-9}{\frac {y(x)}{x}-1}}\right )\right )=c_1+\frac {\log (x)}{2},y(x)\right ],\text {Solve}\left [\frac {1}{8} \left (-\sqrt {\frac {\frac {y(x)}{x}-9}{\frac {y(x)}{x}-1}} \left (\frac {y(x)}{x}-1\right )+\sqrt {\frac {y(x)}{x}-9} \sqrt {\frac {y(x)}{x}-1}+3 \log \left (\frac {y(x)}{x}\right )-\frac {10 \sqrt {\frac {y(x)}{x}-9} \sin ^{-1}\left (\frac {\sqrt {9-\frac {y(x)}{x}}}{2 \sqrt {2}}\right )}{\sqrt {9-\frac {y(x)}{x}}}+6 \tanh ^{-1}\left (\frac {1}{3} \sqrt {\frac {\frac {y(x)}{x}-9}{\frac {y(x)}{x}-1}}\right )+8 \tanh ^{-1}\left (\sqrt {\frac {\frac {y(x)}{x}-9}{\frac {y(x)}{x}-1}}\right )\right )=c_1-\frac {\log (x)}{2},y(x)\right ]\right \}$ Maple : cpu = 0.134 (sec), leaf count = 136

$\left \{ -{\frac {{\it \_C1}}{x} \left ( -y \left ( x \right ) +5\,x+\sqrt {9\,{x}^{2}-10\,xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}} \right ) \left ( {\frac {1}{x} \left ( -y \left ( x \right ) +3\,x+\sqrt {9\,{x}^{2}-10\,xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) ^{-{\frac {3}{2}}}}+x=0,{\frac {{\it \_C1}}{x} \left ( y \left ( x \right ) -5\,x+\sqrt {9\,{x}^{2}-10\,xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}} \right ) \left ( {\frac {1}{x} \left ( -2\,y \left ( x \right ) +6\,x-2\,\sqrt {9\,{x}^{2}-10\,xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) ^{-{\frac {3}{2}}}}+x=0,y \left ( x \right ) =x \right \}$