#### 2.469   ODE No. 469

$a x y'(x)+b y(x)+y(x) y'(x)^2=0$ Mathematica : cpu = 0.278258 (sec), leaf count = 157

$\left \{\text {Solve}\left [\frac {(a+2 b) \log \left (-\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a+2 b\right )+a \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a\right )}{4 (a+b)}=c_1-\frac {\log (x)}{2},y(x)\right ],\text {Solve}\left [\frac {a \log \left (a-\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}\right )+(a+2 b) \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a+2 b\right )}{4 (a+b)}=c_1-\frac {\log (x)}{2},y(x)\right ]\right \}$ Maple : cpu = 0.25 (sec), leaf count = 264

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