ODE No. 1498

2.1498 ODE No. 1498

$(a x^{2} + 6 n) y^{'} (x) - 2 a x y (x) - 2 (n + 1) x y^{″} (x) + x^{2} y^{(3)} (x) = 0$ ✓ Mathematica : cpu = 8.35585 (sec), leaf count = 584

${{y (x) \to - \frac{π c_{3} 2^{- n - \frac{3}{2}} x {(\sqrt{a} x)}^{- n - \frac{1}{2}} (- a^{3 / 2} 2^{2 n} x^{3} \sec (π n) Γ (\frac{3}{2} - n) Γ (n + \frac{3}{2}) J_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x)_{1} {\tilde{F}}_{2} (\frac{3}{2} - n; \frac{1}{2} - n, \frac{5}{2} - n; - \frac{a x^{2}}{4}) - 8 n^{2} {(\sqrt{a} x)}^{2 n} Y_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) - 8 n^{2} \tan (π n) {(\sqrt{a} x)}^{2 n} J_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + a 2^{n + \frac{1}{2}} x^{2} \tan (π n) Γ (n + \frac{3}{2}) {(\sqrt{a} x)}^{n + \frac{1}{2}} J_{n - \frac{1}{2}} (\sqrt{a} x) J_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + a 2^{n + \frac{1}{2}} x^{2} Γ (n + \frac{3}{2}) {(\sqrt{a} x)}^{n + \frac{1}{2}} J_{n - \frac{1}{2}} (\sqrt{a} x) Y_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + 2 {(\sqrt{a} x)}^{2 n} Y_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + 2 \tan (π n) {(\sqrt{a} x)}^{2 n} J_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + \sqrt{a} 2^{n + \frac{3}{2}} x \tan (π n) Γ (n + \frac{3}{2}) {(\sqrt{a} x)}^{n + \frac{1}{2}} J_{n - \frac{3}{2}} (\sqrt{a} x) J_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + \sqrt{a} 2^{n + \frac{3}{2}} x Γ (n + \frac{3}{2}) {(\sqrt{a} x)}^{n + \frac{1}{2}} J_{n - \frac{3}{2}} (\sqrt{a} x) Y_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x))}{a^{3 / 2} Γ (n + \frac{3}{2})} + c_{1} x^{\frac{1}{2} (2 n + 1)} J_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x) + c_{2} x^{\frac{1}{2} (2 n + 1)} Y_{\frac{1}{2} (2 n + 1)} (\sqrt{a} x)}}$

✓ Maple : cpu = 0.237 (sec), leaf count = 53

${y (x) =_C 1 x^{n + \frac{1}{2}} J_{- n - \frac{1}{2}} (\sqrt{a} x) +_C 2 x^{n + \frac{1}{2}} Y_{- n - \frac{1}{2}} (\sqrt{a} x) +_C 3 (a x^{2} + 4 n - 2)}$

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