#### 2.586   ODE No. 586

$y'(x)=\frac {x F\left (\frac {y(x)}{\sqrt {x^2+1}}\right )}{\sqrt {x^2+1}}$ Mathematica : cpu = 0.537654 (sec), leaf count = 975

$\text {Solve}\left [\int _1^x\left (-\frac {K[1] \sqrt {K[1]^2+1} F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^3}{y(x) \left (K[1]^2 F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2-y(x)^2\right )}-\frac {K[1] F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2}{K[1]^2 F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )^2-y(x)^2}+\frac {K[1] F\left (\frac {y(x)}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1} y(x)}\right )dK[1]+\int _1^{y(x)}\left (-\frac {\sqrt {x^2+1} F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )}{-x^2 F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2-F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2+K[2]^2}-\int _1^x\left (\frac {K[1] \sqrt {K[1]^2+1} \left (\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) K[1]^2}{\sqrt {K[1]^2+1}}-2 K[2]+\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^3}{K[2] \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )^2}+\frac {K[1] \sqrt {K[1]^2+1} F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^3}{K[2]^2 \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )}-\frac {3 K[1] F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2}{K[2] \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )}+\frac {K[1] \left (\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) K[1]^2}{\sqrt {K[1]^2+1}}-2 K[2]+\frac {2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2}{\left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )^2}-\frac {2 K[1] F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right ) F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1} \left (K[1]^2 F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2+F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )^2-K[2]^2\right )}-\frac {K[1] F\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1} K[2]^2}+\frac {K[1] F'\left (\frac {K[2]}{\sqrt {K[1]^2+1}}\right )}{\left (K[1]^2+1\right ) K[2]}\right )dK[1]-\frac {K[2]}{-x^2 F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2-F\left (\frac {K[2]}{\sqrt {x^2+1}}\right )^2+K[2]^2}\right )dK[2]=c_1,y(x)\right ]$ Maple : cpu = 0.356 (sec), leaf count = 39

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