ODE No. 1232

4.232 ODE No. 1232

$(x^{2} - 1) \frac{d^{2}}{d x^{2}} y (x) - n (n + 1) y (x) + \frac{\partial}{\partial x} L e g e n d r e P (n, x) = 0$

Mathematica: cpu = 0 (sec), leaf count = 0 $Hanged$

Maple: cpu = 0.141 (sec), leaf count = 418 ${y (x) = (- x^{2} + 1)_{2} F_{1} (\frac{n}{2} + 1, - \frac{n}{2} + \frac{1}{2}; \frac{1}{2}; x^{2})_C 2 + (- x^{3} + x)_{2} F_{1} (- \frac{n}{2} + 1, \frac{n}{2} + \frac{3}{2}; \frac{3}{2}; x^{2})_C 1 - 3 (n + 1) (x - 1) (1 + x) (- \int 1 / 3 \frac{_{2} F_{1} (n / 2 + 1, - n / 2 + 1 / 2; 1 / 2; x^{2}) (x L e g e n d r e P (n, x) - L e g e n d r e P (n + 1, x))}{((_{2} F_{1} (n / 2 + 1, - n / 2 + 1 / 2; 1 / 2; x^{2}) + (n^{2} + n - 2) x^{2}_{2} F_{1} (n / 2 + 2, 3 / 2 - n / 2; 3 / 2; x^{2}))_{2} F_{1} (- n / 2 + 1, n / 2 + 3 / 2; 3 / 2; x^{2}) - 1 / 3_{2} F_{1} (n / 2 + 1, - n / 2 + 1 / 2; 1 / 2; x^{2})_{2} F_{1} (- n / 2 + 2, n / 2 + 5 / 2; 5 / 2; x^{2}) x^{2} (n + 3) (n - 2)) {(x - 1)}^{3} {(1 + x)}^{3}} d x_{2} F_{1} (- n / 2 + 1, n / 2 + 3 / 2; 3 / 2; x^{2}) x + \int 1 / 3 \frac{_{2} F_{1} (- n / 2 + 1, n / 2 + 3 / 2; 3 / 2; x^{2}) x (x L e g e n d r e P (n, x) - L e g e n d r e P (n + 1, x))}{((_{2} F_{1} (n / 2 + 1, - n / 2 + 1 / 2; 1 / 2; x^{2}) + (n^{2} + n - 2) x^{2}_{2} F_{1} (n / 2 + 2, 3 / 2 - n / 2; 3 / 2; x^{2}))_{2} F_{1} (- n / 2 + 1, n / 2 + 3 / 2; 3 / 2; x^{2}) - 1 / 3_{2} F_{1} (n / 2 + 1, - n / 2 + 1 / 2; 1 / 2; x^{2})_{2} F_{1} (- n / 2 + 2, n / 2 + 5 / 2; 5 / 2; x^{2}) x^{2} (n + 3) (n - 2)) {(x - 1)}^{3} {(1 + x)}^{3}} d x_{2} F_{1} (n / 2 + 1, - n / 2 + 1 / 2; 1 / 2; x^{2}))}$

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