#### 2.312   ODE No. 312

$\left (y(x) y'(x)+x\right ) \left (\frac {x^2}{a}+\frac {y(x)^2}{b}\right )+\frac {(a-b) \left (y(x) y'(x)-x\right )}{a+b}=0$ Mathematica : cpu = 0.609407 (sec), leaf count = 204

$\left \{\left \{y(x)\to -\frac {\sqrt {b} \sqrt {2 a^2 W\left (\frac {c_1 (a+b) e^{\frac {b x^2}{2 a^2}-\frac {b}{2 a}-\frac {x^2}{2 b}-\frac {1}{2}}}{2 a^3 b^2}\right )+a^2+a b-a x^2-b x^2}}{\sqrt {a} \sqrt {a+b}}\right \},\left \{y(x)\to \frac {\sqrt {b} \sqrt {2 a^2 W\left (\frac {c_1 (a+b) e^{\frac {b x^2}{2 a^2}-\frac {b}{2 a}-\frac {x^2}{2 b}-\frac {1}{2}}}{2 a^3 b^2}\right )+a^2+a b-a x^2-b x^2}}{\sqrt {a} \sqrt {a+b}}\right \}\right \}$ Maple : cpu = 1.06 (sec), leaf count = 240

$\left \{ y \left ( x \right ) ={\frac {1}{a}\sqrt {a \left ( {{\rm e}^{{\frac {1}{2\,{a}^{2}b} \left ( -2\,{\it lambertW} \left ( 1/2\,{\frac { \left ( a+b \right ) {{\rm e}^{-1/2}}}{{a}^{2}b}{{\rm e}^{-1/2\,{\frac {{x}^{2}}{b}}}}{{\rm e}^{1/2\,{\frac {b{x}^{2}}{{a}^{2}}}}}{{\rm e}^{-1/2\,{\frac {b}{a}}}} \left ( {{\rm e}^{{\frac {{\it \_C1}}{ab}}}} \right ) ^{-1}} \right ) {a}^{2}b+ \left ( -{x}^{2}-b \right ) {a}^{2}+ \left ( -{b}^{2}-2\,{\it \_C1} \right ) a+{b}^{2}{x}^{2} \right ) }}}+b \left ( -{x}^{2}+a \right ) \right ) }},y \left ( x \right ) =-{\frac {1}{a}\sqrt {a \left ( {{\rm e}^{{\frac {1}{2\,{a}^{2}b} \left ( -2\,{\it lambertW} \left ( 1/2\,{\frac { \left ( a+b \right ) {{\rm e}^{-1/2}}}{{a}^{2}b}{{\rm e}^{-1/2\,{\frac {{x}^{2}}{b}}}}{{\rm e}^{1/2\,{\frac {b{x}^{2}}{{a}^{2}}}}}{{\rm e}^{-1/2\,{\frac {b}{a}}}} \left ( {{\rm e}^{{\frac {{\it \_C1}}{ab}}}} \right ) ^{-1}} \right ) {a}^{2}b+ \left ( -{x}^{2}-b \right ) {a}^{2}+ \left ( -{b}^{2}-2\,{\it \_C1} \right ) a+{b}^{2}{x}^{2} \right ) }}}+b \left ( -{x}^{2}+a \right ) \right ) }} \right \}$