ODE No. 283

2.283 ODE No. 283

$3 (y (x)^{2} - x^{2}) y^{'} (x) + 2 y (x)^{3} - 6 x (x + 1) y (x) - 3 e^{x} = 0$ ✓ Mathematica : cpu = 0.159563 (sec), leaf count = 477

${{y (x) \to - \frac{e^{- 2 x} \sqrt[3]{\sqrt{(27 c_{1} e^{4 x} - 27 e^{7 x})^{2} - 2916 e^{12 x} x^{6}} + 27 c_{1} e^{4 x} - 27 e^{7 x}}}{3 \sqrt[3]{2}} - \frac{3 \sqrt[3]{2} e^{2 x} x^{2}}{\sqrt[3]{\sqrt{(27 c_{1} e^{4 x} - 27 e^{7 x})^{2} - 2916 e^{12 x} x^{6}} + 27 c_{1} e^{4 x} - 27 e^{7 x}}}}, {y (x) \to \frac{(1 - i \sqrt{3}) e^{- 2 x} \sqrt[3]{\sqrt{(27 c_{1} e^{4 x} - 27 e^{7 x})^{2} - 2916 e^{12 x} x^{6}} + 27 c_{1} e^{4 x} - 27 e^{7 x}}}{6 \sqrt[3]{2}} + \frac{3 (1 + i \sqrt{3}) e^{2 x} x^{2}}{2^{2 / 3} \sqrt[3]{\sqrt{(27 c_{1} e^{4 x} - 27 e^{7 x})^{2} - 2916 e^{12 x} x^{6}} + 27 c_{1} e^{4 x} - 27 e^{7 x}}}}, {y (x) \to \frac{(1 + i \sqrt{3}) e^{- 2 x} \sqrt[3]{\sqrt{(27 c_{1} e^{4 x} - 27 e^{7 x})^{2} - 2916 e^{12 x} x^{6}} + 27 c_{1} e^{4 x} - 27 e^{7 x}}}{6 \sqrt[3]{2}} + \frac{3 (1 - i \sqrt{3}) e^{2 x} x^{2}}{2^{2 / 3} \sqrt[3]{\sqrt{(27 c_{1} e^{4 x} - 27 e^{7 x})^{2} - 2916 e^{12 x} x^{6}} + 27 c_{1} e^{4 x} - 27 e^{7 x}}}}}$

✓ Maple : cpu = 0.069 (sec), leaf count = 622

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

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