- #1
member 428835
Hi PF!
I'm wondering what the fundamental solution is for this ODE
$$
f''(x)+\cot (x) f'(x) + \left( 2-\frac{L^2}{\sin(x)} \right) f(x) = 0 : L = 2,3,4...
$$
I know one solution is $$
(\cos(x)+L)\left(\frac{1-\cos(s)}{1+\cos(s)}\right)^{L/2}
$$
but I don't know the other. Mathematica isn't much help here, as it only gives me a solution but not for ##L=1## (Legendre Polynomials) but not a general solution for ##L\geq 2##. Any help is very appreciated!
I'm wondering what the fundamental solution is for this ODE
$$
f''(x)+\cot (x) f'(x) + \left( 2-\frac{L^2}{\sin(x)} \right) f(x) = 0 : L = 2,3,4...
$$
I know one solution is $$
(\cos(x)+L)\left(\frac{1-\cos(s)}{1+\cos(s)}\right)^{L/2}
$$
but I don't know the other. Mathematica isn't much help here, as it only gives me a solution but not for ##L=1## (Legendre Polynomials) but not a general solution for ##L\geq 2##. Any help is very appreciated!