The Legendre polynomials are the solutions of an important differential equation in Science, namely $$ \begin{equation*} C(1-x^2)P-m_l^2P+(1-x^2)\frac{d}{dx}\left((1-x^2)\frac{dP}{dx}\right)=0. \end{equation*} $$ Here \( C \) is a constant. For \( m_l=0 \) we obtain the Legendre polynomials as solutions, whereas \( m_l \ne 0 \) yields the so-called associated Legendre polynomials. This differential equation arises in for example the solution of the angular dependence of Schroedinger's equation with spherically symmetric potentials such as the Coulomb potential.