= Solution
Legendre's equation is
$$
-\frac d{dx}\left((1-x^2)y'\right)=\lambda y.
$$
With $\langle f,g\rangle=\int_{-1}^1f\bar g\,dx$, 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 $y=\sum a_nx^n$ gives
$$
\frac{a_{n+2}}{a_n}=\frac{n(n+1)-\lambda}{(n+1)(n+2)}.
$$
The <series> terminates at degree $\ell$ when $\lambda=\ell(\ell+1)$. Normalizing at one gives
$$
\boxed{P_1(x)=x,\qquad P_3(x)=\frac12(5x^3-3x).}
$$
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