ODE No. 1189

4.189 ODE No. 1189

$x^{2} \frac{d^{2}}{d x^{2}} y (x) + a x \frac{d}{d x} y (x) + (b x^{m} + c) y (x) = 0$

Mathematica: cpu = 0.069509 (sec), leaf count = 445 ${{y (x) \to c_{1} m^{- \frac{- \sqrt{a^{2} - 2 a - 4 c + 1} - a + 1}{m} - \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{m}} b^{\frac{- \sqrt{a^{2} - 2 a - 4 c + 1} - a + 1}{2 m} + \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{2 m}} {(x^{m})}^{\frac{- \sqrt{a^{2} - 2 a - 4 c + 1} - a + 1}{2 m} + \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{2 m}} Γ (1 - \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{m}) J_{- \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{m}} (\frac{2 \sqrt{b} \sqrt{x^{m}}}{m}) + c_{2} m^{\frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{m} - \frac{\sqrt{a^{2} - 2 a - 4 c + 1} - a + 1}{m}} b^{\frac{\sqrt{a^{2} - 2 a - 4 c + 1} - a + 1}{2 m} - \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{2 m}} {(x^{m})}^{\frac{\sqrt{a^{2} - 2 a - 4 c + 1} - a + 1}{2 m} - \frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{2 m}} Γ (\frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{m} + 1) J_{\frac{\sqrt{a^{2} - 2 a - 4 c + 1}}{m}} (\frac{2 \sqrt{b} \sqrt{x^{m}}}{m})}}$

Maple: cpu = 0.031 (sec), leaf count = 85 ${y (x) =_C 1 x^{- \frac{a}{2} + \frac{1}{2}} J_{\frac{1}{m} \sqrt{a^{2} - 2 a - 4 c + 1}} (2 \frac{\sqrt{b} x^{m / 2}}{m}) +_C 2 x^{- \frac{a}{2} + \frac{1}{2}} Y_{\frac{1}{m} \sqrt{a^{2} - 2 a - 4 c + 1}} (2 \frac{\sqrt{b} x^{m / 2}}{m})}$

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