ODE No. 420

2.420 ODE No. 420

Problem in Latex
Mathematica input
Maple input

$a + x y^{'} (x)^{2} - 2 y (x) y^{'} (x) = 0$ ✓ Mathematica : cpu = 0.281039 (sec), leaf count = 777

${{y (x) \to \frac{1}{4} a^{2} e^{- \frac{3 c_{1}}{2}} x^{2} + \frac{1}{4} e^{- \frac{3 c_{1}}{2}} \sqrt[3]{a^{6} x^{6} - 20 a^{3} e^{3 c_{1}} x^{3} + 8 \sqrt{- a^{9} e^{3 c_{1}} x^{9} + 3 a^{6} e^{6 c_{1}} x^{6} - 3 a^{3} e^{9 c_{1}} x^{3} + e^{12 c_{1}}} - 8 e^{6 c_{1}}} - \frac{e^{- \frac{3 c_{1}}{2}} (- 9 a^{4} x^{4} - 72 a e^{3 c_{1}} x)}{36 \sqrt[3]{a^{6} x^{6} - 20 a^{3} e^{3 c_{1}} x^{3} + 8 \sqrt{- a^{9} e^{3 c_{1}} x^{9} + 3 a^{6} e^{6 c_{1}} x^{6} - 3 a^{3} e^{9 c_{1}} x^{3} + e^{12 c_{1}}} - 8 e^{6 c_{1}}}}}, {y (x) \to \frac{1}{4} a^{2} e^{- \frac{3 c_{1}}{2}} x^{2} - \frac{1}{8} (1 - i \sqrt{3}) e^{- \frac{3 c_{1}}{2}} \sqrt[3]{a^{6} x^{6} - 20 a^{3} e^{3 c_{1}} x^{3} + 8 \sqrt{- a^{9} e^{3 c_{1}} x^{9} + 3 a^{6} e^{6 c_{1}} x^{6} - 3 a^{3} e^{9 c_{1}} x^{3} + e^{12 c_{1}}} - 8 e^{6 c_{1}}} + \frac{(1 + i \sqrt{3}) e^{- \frac{3 c_{1}}{2}} (- 9 a^{4} x^{4} - 72 a e^{3 c_{1}} x)}{72 \sqrt[3]{a^{6} x^{6} - 20 a^{3} e^{3 c_{1}} x^{3} + 8 \sqrt{- a^{9} e^{3 c_{1}} x^{9} + 3 a^{6} e^{6 c_{1}} x^{6} - 3 a^{3} e^{9 c_{1}} x^{3} + e^{12 c_{1}}} - 8 e^{6 c_{1}}}}}, {y (x) \to \frac{1}{4} a^{2} e^{- \frac{3 c_{1}}{2}} x^{2} - \frac{1}{8} (1 + i \sqrt{3}) e^{- \frac{3 c_{1}}{2}} \sqrt[3]{a^{6} x^{6} - 20 a^{3} e^{3 c_{1}} x^{3} + 8 \sqrt{- a^{9} e^{3 c_{1}} x^{9} + 3 a^{6} e^{6 c_{1}} x^{6} - 3 a^{3} e^{9 c_{1}} x^{3} + e^{12 c_{1}}} - 8 e^{6 c_{1}}} + \frac{(1 - i \sqrt{3}) e^{- \frac{3 c_{1}}{2}} (- 9 a^{4} x^{4} - 72 a e^{3 c_{1}} x)}{72 \sqrt[3]{a^{6} x^{6} - 20 a^{3} e^{3 c_{1}} x^{3} + 8 \sqrt{- a^{9} e^{3 c_{1}} x^{9} + 3 a^{6} e^{6 c_{1}} x^{6} - 3 a^{3} e^{9 c_{1}} x^{3} + e^{12 c_{1}}} - 8 e^{6 c_{1}}}}}}$ ✓ Maple : cpu = 0.189 (sec), leaf count = 689

${y (x) = \frac{x}{12_C 1} (4 \frac{x^{2}}{\sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}} + 2 x + \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}) + 3_C 1 a {(4 \frac{x^{2}}{\sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}} + 2 x + \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1})}^{- 1}, y (x) = \frac{x}{6_C 1} ((- 1 + i \sqrt{3}) x^{2} \frac{1}{\sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}} - \frac{i}{4} \sqrt{3} \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1} + x - \frac{1}{4} \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}) + 6_C 1 a {(4 \frac{(- 1 + i \sqrt{3}) x^{2}}{\sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}} - i \sqrt{3} \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1} + 4 x - \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1})}^{- 1}, y (x) = - \frac{x}{6_C 1} ((1 + i \sqrt{3}) x^{2} \frac{1}{\sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}} - \frac{i}{4} \sqrt{3} \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1} - x + \frac{1}{4} \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}) - 6_C 1 a {(4 \frac{(1 + i \sqrt{3}) x^{2}}{\sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1}} - i \sqrt{3} \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1} - 4 x + \sqrt[3]{- 36 a {_C 1}^{2} + 8 x^{3} + 12 \sqrt{a (9 a {_C 1}^{2} - 4 x^{3})}_C 1})}^{- 1}}$

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