ODE No. 157

2.157 ODE No. 157

$a (y (x)^{2} - 2 x y (x) + 1) + (x^{2} - 1) y^{'} (x) = 0$ ✓ Mathematica : cpu = 0.0858136 (sec), leaf count = 158

${{y (x) \to \frac{(x^{2} - 1) (c_{1} (a x {(x^{2} - 1)}^{\frac{a}{2} - 1} P_{a - 1} (x) + {(x^{2} - 1)}^{\frac{a}{2} - 1} (a P_{a} (x) - a x P_{a - 1} (x))) + a x {(x^{2} - 1)}^{\frac{a}{2} - 1} Q_{a - 1} (x) + {(x^{2} - 1)}^{\frac{a}{2} - 1} (a Q_{a} (x) - a x Q_{a - 1} (x)))}{a (c_{1} {(x^{2} - 1)}^{a / 2} P_{a - 1} (x) + {(x^{2} - 1)}^{a / 2} Q_{a - 1} (x))}}}$

✓ Maple : cpu = 0.266 (sec), leaf count = 231

${y (x) = \frac{1}{4 a (1 + x)} (8_C 1 (1 + x) ((a - 1 / 2) x - a / 2 + 1 / 2) H e u n C (0, - 2 a + 1, 0, 0, a^{2} - a + 1 / 2, 2 {(1 + x)}^{- 1}) - a {(- \frac{x}{2} - \frac{1}{2})}^{- 2 a + 1} (1 + x) H e u n C (0, 2 a - 1, 0, 0, a^{2} - a + \frac{1}{2}, 2 {(1 + x)}^{- 1}) - 8 (x - 1) (H e u n C P r i m e (0, - 2 a + 1, 0, 0, a^{2} - a + 1 / 2, 2 {(1 + x)}^{- 1})_C 1 - 1 / 4 {(- x / 2 - 1 / 2)}^{- 2 a + 1} H e u n C P r i m e (0, 2 a - 1, 0, 0, a^{2} - a + 1 / 2, 2 {(1 + x)}^{- 1}))) {(H e u n C (0, - 2 a + 1, 0, 0, a^{2} - a + \frac{1}{2}, 2 {(1 + x)}^{- 1})_C 1 - \frac{1}{4} {(- \frac{x}{2} - \frac{1}{2})}^{- 2 a + 1} H e u n C (0, 2 a - 1, 0, 0, a^{2} - a + \frac{1}{2}, 2 {(1 + x)}^{- 1}))}^{- 1}}$

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