ODE No. 1564

6.31 ODE No. 1564

$x^{4} d 4 y (x) + 4 x^{3} \frac{d^{3}}{d x^{3}} y (x) - (4 n^{2} + 3) x^{2} \frac{d^{2}}{d x^{2}} y (x) + (12 n^{2} - 3) x \frac{d}{d x} y (x) - (4 x^{4} + 12 n^{2} - 3) y (x) = 0$

Mathematica: cpu = 1.386176 (sec), leaf count = 230 ${{y (x) \to \frac{\sqrt[4]{- 1} c_{1} x_{0} F_{3} (; \frac{1}{2}, \frac{3}{2} - \frac{n}{2}, \frac{n}{2} + \frac{3}{2}; \frac{x^{4}}{64})}{2 \sqrt{2}} + c_{3} (- 1)^{\frac{1}{4} (- 2 n - 1)} 2^{2 n + \frac{1}{2} (2 n + 1) + 1} x^{- 2 n - 1}_{0} F_{3} (; 1 - n, \frac{1}{2} - \frac{n}{2}, - \frac{n}{2}; \frac{x^{4}}{64}) + c_{4} (- 1)^{\frac{1}{4} (2 n - 1)} 2^{\frac{1}{2} (1 - 2 n) - 2 n + 1} x^{2 n - 1}_{0} F_{3} (; \frac{n}{2} + \frac{1}{2}, \frac{n}{2}, n + 1; \frac{x^{4}}{64}) + \frac{(- 1)^{3 / 4} c_{2} x^{3}_{0} F_{3} (; \frac{3}{2}, 2 - \frac{n}{2}, \frac{n}{2} + 2; \frac{x^{4}}{64})}{16 \sqrt{2}}}}$

Maple: cpu = 0.156 (sec), leaf count = 156 ${y (x) = \frac{_C 1 ({({b e r}_{n} (x))}^{2} + {({b e i}_{n} (x))}^{2})}{x} +_C 2 x^{3}_{0} F_{3} (; \frac{3}{2}, 2 - \frac{n}{2}, 2 + \frac{n}{2}; \frac{x^{4}}{64}) +_C 3 x_{0} F_{3} (; \frac{1}{2}, \frac{3}{2} + \frac{n}{2}, \frac{3}{2} - \frac{n}{2}; \frac{x^{4}}{64}) + \frac{_C 4 (- \sqrt{2} {({b e r}_{- n} (x))}^{2} n - \sqrt{2} {({b e i}_{- n} (x))}^{2} n + {b e r}_{- n} (x) {b e r}_{1 - n} (x) x - {b e r}_{1 - n} (x) {b e i}_{- n} (x) x + {b e r}_{- n} (x) {b e i}_{1 - n} (x) x + {b e i}_{- n} (x) {b e i}_{1 - n} (x) x)}{x}}$

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