Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 13B (Methods)

Legendre's equation isWith $⟨f,g⟩={\int }_{-1}^{1}f\bar{g}dx$, integration by parts has no endpoint term because $1-x^2=0$ there, so the operator is self-adjoint. Sturm–Liouville eigenvalues are real, can be ordered increasingly, and their eigenfunctions are orthogonal and complete under standard regularity assumptions.

Substitution of $y=\sum a_n{x}^n$ givesThe series terminates at degree $\ell$ when $\lambda =\ell (\ell +1)$. Normalizing at one gives

The separated axisymmetric solution isSince $x(1-x^2)=\frac{2}{5}(P_1-P_3)$, regularity at the origin and the boundary condition give