ODE No. 430

2.430 ODE No. 430

Problem in Latex
Mathematica input
Maple input

$a0 x + y^{'} (x) (a1 x + b1 y (x) + c1) + (a2 x + c2) y^{'} (x)^{2} + b0 y (x) + c0 = 0$ ✓ Mathematica : cpu = 238.665 (sec), leaf count = 478

$Solve [{x = (b0 + b1 K $ 379019) \exp (\frac{(b1 (b0 - a1) + 2 a2 b0) \tan^{- 1} (\frac{a1 + 2 K $ 379019 (a2 + b1) + b0}{\sqrt{4 a0 (a2 + b1) - {a1}^{2} - 2 a1 b0 - {b0}^{2}}})}{(a2 + b1) \sqrt{4 a0 (a2 + b1) - {a1}^{2} - 2 a1 b0 - {b0}^{2}}} - \frac{(2 a2 + b1) \log (a0 + K $ 379019 (a1 + K $ 379019 (a2 + b1) + b0))}{2 (a2 + b1)}) \int \frac{(\frac{- c1 - 2 c2 K $ 379019}{b0 + b1 K $ 379019} - \frac{b1 (- c0 - c1 K $ 379019 - c2 {K $ 379019}^{2})}{(b0 + b1 K $ 379019)^{2}}) \exp (\frac{(2 a2 + b1) \log (a0 + K $ 379019 (a1 + K $ 379019 (a2 + b1) + b0))}{2 (a2 + b1)} - \frac{(b1 (b0 - a1) + 2 a2 b0) \tan^{- 1} (\frac{a1 + 2 K $ 379019 (a2 + b1) + b0}{\sqrt{4 a0 (a2 + b1) - {a1}^{2} - 2 a1 b0 - {b0}^{2}}})}{(a2 + b1) \sqrt{4 a0 (a2 + b1) - {a1}^{2} - 2 a1 b0 - {b0}^{2}}})}{(b0 + b1 K $ 379019) (K $ 379019 - \frac{- a0 - a1 K $ 379019 - a2 {K $ 379019}^{2}}{b0 + b1 K $ 379019})} d K $ 379019 + c_{1} (b0 + b1 K $ 379019) \exp (\frac{(b1 (b0 - a1) + 2 a2 b0) \tan^{- 1} (\frac{a1 + 2 K $ 379019 (a2 + b1) + b0}{\sqrt{4 a0 (a2 + b1) - {a1}^{2} - 2 a1 b0 - {b0}^{2}}})}{(a2 + b1) \sqrt{4 a0 (a2 + b1) - {a1}^{2} - 2 a1 b0 - {b0}^{2}}} - \frac{(2 a2 + b1) \log (a0 + K $ 379019 (a1 + K $ 379019 (a2 + b1) + b0))}{2 (a2 + b1)}), y (x) = \frac{x (- a0 - a1 K $ 379019 - a2 {K $ 379019}^{2})}{b0 + b1 K $ 379019} + \frac{- c0 - c1 K $ 379019 - c2 {K $ 379019}^{2}}{b0 + b1 K $ 379019}}, {y (x), K $ 379019}]$ ✓ Maple : cpu = 1.861 (sec), leaf count = 1602

${\frac{1}{2 a 2 x + 2 c 2} (2 e^{\frac{- 2 a 2 b 0 + b 1 (a 1 - b 0)}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}} (a 2 + b 1)} \arctan (\frac{(a 2 + b 1) \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} + b 1 (a 2 + b 1) y (x) + (- b 0 x + c 1) a 2 + (a 1 x + c 1) b 1 - c 2 (a 1 + b 0)}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}} (a 2 x + c 2)})} (b 1 \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} + {b 1}^{2} y (x) + (a 1 x + c 1) b 1 - 2 b 0 (a 2 x + c 2)) (_C 1 + 1 / 2 \int^{\frac{- a 1 x - b 1 y (x) - \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} - c 1}{2 a 2 x + 2 c 2}} - \frac{(2_a c 2 + c 1) b 0 + b 1 ({_a}^{2} c 2 - c 0)}{{(_a b 1 + b 0)}^{2}} e^{\frac{- 2 a 2 b 0 + b 1 (a 1 - b 0)}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}} (a 2 + b 1)} \arctan (\frac{(2 a 2 + 2 b 1)_a + a 1 + b 0}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}}})} {((a 2 + b 1) {_a}^{2} + (a 1 + b 0)_a + a 0)}^{- \frac{b 1}{2 a 2 + 2 b 1}} d_a) {(1 / 2 \frac{(b 1 (a 2 + b 1) y (x) + (- b 0 x + c 1) a 2 + (a 1 x + c 1) b 1 - c 2 (a 1 + b 0)) \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} + {b 1}^{2} (a 2 + b 1) {(y (x))}^{2} + ((2 a 1 x + 2 c 1) {b 1}^{2} + (((a 1 - 3 b 0) x + 2 c 1) a 2 - c 2 (a 1 + 3 b 0)) b 1 - 2 b 0 a 2 (a 2 x + c 2)) y (x) + ((- 2 a 0 x^{2} - 2 c 0 x) a 2 + {a 1}^{2} x^{2} + (- 2 a 0 c 2 + 2 c 1 a 1) x - 2 c 0 c 2 + {c 1}^{2}) b 1 - 2 {a 2}^{2} c 0 x + (- a 1 b 0 x^{2} + (2 a 0 c 2 + c 1 (a 1 - b 0)) x - 2 c 0 c 2 + {c 1}^{2}) a 2 - (a 1 (a 1 + b 0) x - 2 a 0 c 2 + c 1 (a 1 + b 0)) c 2}{{(a 2 x + c 2)}^{2}})}^{\frac{- 2 a 2 - b 1}{2 a 2 + 2 b 1}} + 2 a 2 x^{2} + 2 x c 2) = 0, \frac{1}{2 a 2 x + 2 c 2} (- 2 e^{\frac{- 2 a 2 b 0 + b 1 (a 1 - b 0)}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}} (a 2 + b 1)} \arctan (\frac{(- a 2 - b 1) \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} + b 1 (a 2 + b 1) y (x) + (- b 0 x + c 1) a 2 + (a 1 x + c 1) b 1 - c 2 (a 1 + b 0)}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}} (a 2 x + c 2)})} (- b 1 \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} + {b 1}^{2} y (x) + (a 1 x + c 1) b 1 - 2 b 0 (a 2 x + c 2)) (_C 1 - 1 / 2 \int^{\frac{- a 1 x - b 1 y (x) - c 1 + \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}}}{2 a 2 x + 2 c 2}} - \frac{(2_a c 2 + c 1) b 0 + b 1 ({_a}^{2} c 2 - c 0)}{{(_a b 1 + b 0)}^{2}} e^{\frac{- 2 a 2 b 0 + b 1 (a 1 - b 0)}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}} (a 2 + b 1)} \arctan (\frac{(2 a 2 + 2 b 1)_a + a 1 + b 0}{\sqrt{(4 a 2 + 4 b 1) a 0 - {(a 1 + b 0)}^{2}}})} {((a 2 + b 1) {_a}^{2} + (a 1 + b 0)_a + a 0)}^{- \frac{b 1}{2 a 2 + 2 b 1}} d_a) {(1 / 2 \frac{(- b 1 (a 2 + b 1) y (x) + (- a 1 x - c 1) b 1 + (b 0 x - c 1) a 2 + c 2 (a 1 + b 0)) \sqrt{{b 1}^{2} {(y (x))}^{2} + ((2 b 1 a 1 - 4 a 2 b 0) x - 4 b 0 c 2 + 2 b 1 c 1) y (x) + (- 4 a 0 a 2 + {a 1}^{2}) x^{2} + (- 4 a 0 c 2 + 2 c 1 a 1 - 4 a 2 c 0) x - 4 c 0 c 2 + {c 1}^{2}} + {b 1}^{2} (a 2 + b 1) {(y (x))}^{2} + ((2 a 1 x + 2 c 1) {b 1}^{2} + (((a 1 - 3 b 0) x + 2 c 1) a 2 - c 2 (a 1 + 3 b 0)) b 1 - 2 b 0 a 2 (a 2 x + c 2)) y (x) + ((- 2 a 0 x^{2} - 2 c 0 x) a 2 + {a 1}^{2} x^{2} + (- 2 a 0 c 2 + 2 c 1 a 1) x - 2 c 0 c 2 + {c 1}^{2}) b 1 - 2 {a 2}^{2} c 0 x + (- a 1 b 0 x^{2} + (2 a 0 c 2 + c 1 (a 1 - b 0)) x - 2 c 0 c 2 + {c 1}^{2}) a 2 - (a 1 (a 1 + b 0) x - 2 a 0 c 2 + c 1 (a 1 + b 0)) c 2}{{(a 2 x + c 2)}^{2}})}^{\frac{- 2 a 2 - b 1}{2 a 2 + 2 b 1}} + 2 a 2 x^{2} + 2 x c 2) = 0}$

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