#### 2.579   ODE No. 579

$y'(x)=F\left (\frac {a x^2}{4}+\frac {b x}{2}+y(x)\right )-\frac {a x}{2}$ Mathematica : cpu = 0.247811 (sec), leaf count = 514

$\text {Solve}\left [\int _1^{y(x)}-\frac {b \int _1^x\left (\frac {2 a K[1] F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right ) \int _1^x\left (\frac {2 a K[1] F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2}{b+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right )}dK[2]+\int _1^x\left (\frac {2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}-\frac {a K[1]}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}\right )dK[1]=c_1,y(x)\right ]$ Maple : cpu = 0.18 (sec), leaf count = 35

$\left \{ y \left ( x \right ) =-{\frac {a{x}^{2}}{4}}-{\frac {bx}{2}}+{\it RootOf} \left ( -x+2\,\int ^{{\it \_Z}}\! \left ( 2\,F \left ( {\it \_a} \right ) +b \right ) ^{-1}{d{\it \_a}}+{\it \_C1} \right ) \right \}$