I am a physics undergraduate. I am consciously and specifically asking this question in math stack exchange and not physics stack exchange. I am working on solving for electric potential in charge free regions via a boundary value approach, i.e., solving Laplace's equation.
$$\nabla^2\Phi=0.$$
The solutions are spherical harmonics. In these solutions are the Legendre polynomials with $cos\theta$ substituted for $x$.
I.e. if
$$P_l(x) = 1 + x + \frac{1}{2}(3x^2-1) + ...$$
Then,
$$P_l(cos\theta) = 1 + cos\theta+\frac{1}{2}(3cos^2\theta-1)+...$$
My Question: Is there general way to write:
$$\frac{\partial^2}{\partial\theta}(P_l(cos\theta)).$$
I am actually trying to solve:
$$CP_l(cos\theta) = -\frac{\partial^2}{\partial\theta}(P_l(cos\theta))$$
Where C is a constant. It looks very much like a differential equation whose solution is sin/cos, but I think it is maybe a superficial similarity.
Any help would be greatly appreciated!