ODE No. 579

3.579 ODE No. 579

$\frac{d}{d x} y (x) = - 1 / 2 a x + F (y (x) + 1 / 4 a x^{2} + 1 / 2 b x) = 0$

Mathematica: cpu = 13.258184 (sec), leaf count = 510 $Solve [\int_{1}^{y (x)} - \frac{2 F (K [2] + \frac{a x^{2}}{4} + \frac{b x}{2}) \int_{1}^{x} (\frac{2 a K [1] F^{'} (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2])}{{(2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) + b)}^{2}} + \frac{2 F^{'} (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2])}{2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) + b} - \frac{4 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) F^{'} (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2])}{{(2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) + b)}^{2}}) d K [1] + b \int_{1}^{x} (\frac{2 a K [1] F^{'} (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2])}{{(2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) + b)}^{2}} + \frac{2 F^{'} (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2])}{2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) + b} - \frac{4 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) F^{'} (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2])}{{(2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + K [2]) + b)}^{2}}) d K [1] + 2}{2 F (K [2] + \frac{a x^{2}}{4} + \frac{b x}{2}) + b} d K [2] + \int_{1}^{x} (\frac{2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + y (x))}{2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + y (x)) + b} - \frac{a K [1]}{2 F (\frac{1}{4} a K [1]^{2} + \frac{1}{2} b K [1] + y (x)) + b}) d K [1] = c_{1}, y (x)]$

Maple: cpu = 0.031 (sec), leaf count = 35 ${y (x) = - \frac{a x^{2}}{4} - \frac{b x}{2} + R o o t O f (- x + 2 \int^{_Z} {(2 F (_a) + b)}^{- 1} d_a +_C 1)}$

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