Solution to Legendre's Differential Equation

Equation
Legendre's differential equation is a second order ODE of the form:
 * $$\left({1 - x^2}\right) \frac{\mathrm d^2 y} {\mathrm d x^2} - 2 x \frac{\mathrm d y} {\mathrm d x} + p \left({p + 1}\right) y = 0$$

The parameter $$p$$ may be any arbitrary real or complex number.

Solution
The solutions of Legendre's differential equation are known as Legendre polynomials, and they are functions of the parameter $$p$$.