#### 2.591   ODE No. 591

$y'(x)=\frac {x F\left (\frac {a y(x)^2+b x^2}{a}\right )}{\sqrt {a} y(x)}$ Mathematica : cpu = 0.456118 (sec), leaf count = 253

$\text {Solve}\left [\int _1^{y(x)}\left (-\frac {b K[2]}{b+\sqrt {a} F\left (\frac {b x^2+a K[2]^2}{a}\right )}-\int _1^x\left (\frac {2 b K[1] K[2] F'\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )}-\frac {2 b F\left (\frac {b K[1]^2+a K[2]^2}{a}\right ) K[1] K[2] F'\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b F\left (\frac {b K[1]^2+a y(x)^2}{a}\right ) K[1]}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )\right )}dK[1]=c_1,y(x)\right ]$ Maple : cpu = 0.565 (sec), leaf count = 108

$\left \{ y \left ( x \right ) ={\frac {1}{a}\sqrt {a \left ( -b{x}^{2}+{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) a+b\sqrt {a} \right ) ^{-1}{d{\it \_a}}b{a}^{{\frac {3}{2}}}-b{x}^{2}+2\,{\it \_C1}\,a \right ) a \right ) }},y \left ( x \right ) =-{\frac {1}{a}\sqrt {a \left ( -b{x}^{2}+{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) a+b\sqrt {a} \right ) ^{-1}{d{\it \_a}}b{a}^{{\frac {3}{2}}}-b{x}^{2}+2\,{\it \_C1}\,a \right ) a \right ) }} \right \}$