ODE No. 1410

2.1410 ODE No. 1410

$y^{″} (x) = - \frac{y^{'} (x) (a p x^{b} + q)}{x (a x^{b} - 1)} - \frac{y (x) (a r x^{b} + s)}{x^{2} (a x^{b} - 1)}$ ✓ Mathematica : cpu = 0.134248 (sec), leaf count = 481

${{y (x) \to c_{1} i^{\frac{- \sqrt{q^{2} + 2 q + 4 s + 1} + q + 1}{b}} a^{\frac{- \sqrt{q^{2} + 2 q + 4 s + 1} + q + 1}{2 b}} {(x^{b})}^{\frac{- \sqrt{q^{2} + 2 q + 4 s + 1} + q + 1}{2 b}}_{2} F_{1} (\frac{p}{2 b} + \frac{q}{2 b} - \frac{\sqrt{p^{2} - 2 p - 4 r + 1}}{2 b} - \frac{\sqrt{q^{2} + 2 q + 4 s + 1}}{2 b}, \frac{p}{2 b} + \frac{q}{2 b} + \frac{\sqrt{p^{2} - 2 p - 4 r + 1}}{2 b} - \frac{\sqrt{q^{2} + 2 q + 4 s + 1}}{2 b}; 1 - \frac{\sqrt{q^{2} + 2 q + 4 s + 1}}{b}; a x^{b}) + c_{2} i^{\frac{\sqrt{q^{2} + 2 q + 4 s + 1} + q + 1}{b}} a^{\frac{\sqrt{q^{2} + 2 q + 4 s + 1} + q + 1}{2 b}} {(x^{b})}^{\frac{\sqrt{q^{2} + 2 q + 4 s + 1} + q + 1}{2 b}}_{2} F_{1} (\frac{p}{2 b} + \frac{q}{2 b} - \frac{\sqrt{p^{2} - 2 p - 4 r + 1}}{2 b} + \frac{\sqrt{q^{2} + 2 q + 4 s + 1}}{2 b}, \frac{p}{2 b} + \frac{q}{2 b} + \frac{\sqrt{p^{2} - 2 p - 4 r + 1}}{2 b} + \frac{\sqrt{q^{2} + 2 q + 4 s + 1}}{2 b}; \frac{\sqrt{q^{2} + 2 q + 4 s + 1}}{b} + 1; a x^{b})}}$

✓ Maple : cpu = 0.276 (sec), leaf count = 253

${y (x) =_C 1_{2} F_{1} (\frac{1}{2 b} (p + q + \sqrt{q^{2} + 2 q + 4 s + 1} + \sqrt{p^{2} - 2 p - 4 r + 1}), \frac{1}{2 b} (p + q + \sqrt{q^{2} + 2 q + 4 s + 1} - \sqrt{p^{2} - 2 p - 4 r + 1}); \frac{1}{b} (b + \sqrt{q^{2} + 2 q + 4 s + 1}); a x^{b}) x^{\frac{q}{2} + \frac{1}{2} \sqrt{q^{2} + 2 q + 4 s + 1} + \frac{1}{2}} +_C 2_{2} F_{1} (- \frac{1}{2 b} (- p - q + \sqrt{q^{2} + 2 q + 4 s + 1} + \sqrt{p^{2} - 2 p - 4 r + 1}), \frac{1}{2 b} (p + q - \sqrt{q^{2} + 2 q + 4 s + 1} + \sqrt{p^{2} - 2 p - 4 r + 1}); \frac{1}{b} (b - \sqrt{q^{2} + 2 q + 4 s + 1}); a x^{b}) x^{\frac{q}{2} - \frac{1}{2} \sqrt{q^{2} + 2 q + 4 s + 1} + \frac{1}{2}}}$

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