#### 2.424   ODE No. 424

$a y(x) y'(x)+b x+x y'(x)^2=0$ Mathematica : cpu = 0.489811 (sec), leaf count = 223

$\left \{\text {Solve}\left [\frac {-2 a \tan ^{-1}\left (\frac {a y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )+(a+2) \left (2 \tan ^{-1}\left (\frac {(a+2) y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )-i \log \left (\frac {(a+1) y(x)^2}{x^2}+b\right )\right )}{8 (a+1)}=c_1+\frac {1}{2} i \log (x),y(x)\right ],\text {Solve}\left [\frac {-2 a \tan ^{-1}\left (\frac {a y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )+(a+2) \left (2 \tan ^{-1}\left (\frac {(a+2) y(x)}{x \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}\right )+i \log \left (\frac {(a+1) y(x)^2}{x^2}+b\right )\right )}{8 (a+1)}=c_1-\frac {1}{2} i \log (x),y(x)\right ]\right \}$ Maple : cpu = 0.285 (sec), leaf count = 193

$\left \{ {\frac {1}{x} \left ( -{\it \_C1}\, \left ( ay \left ( x \right ) -\sqrt {{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,b{x}^{2}} \right ) \left ( {\frac {a}{2\,{x}^{2}} \left ( -y \left ( x \right ) \left ( a+1 \right ) \sqrt {{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,b{x}^{2}}+ \left ( {a}^{2}+a \right ) \left ( y \left ( x \right ) \right ) ^{2}-2\,b{x}^{2} \right ) } \right ) ^{{\frac {-a-2}{2\,a+2}}}+{x}^{2} \right ) }=0,{\frac {1}{x} \left ( {\it \_C1}\, \left ( ay \left ( x \right ) +\sqrt {{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,b{x}^{2}} \right ) \left ( {\frac {a}{2\,{x}^{2}} \left ( y \left ( x \right ) \left ( a+1 \right ) \sqrt {{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,b{x}^{2}}+ \left ( {a}^{2}+a \right ) \left ( y \left ( x \right ) \right ) ^{2}-2\,b{x}^{2} \right ) } \right ) ^{{\frac {-a-2}{2\,a+2}}}+{x}^{2} \right ) }=0 \right \}$