Q: Solve y''+λy = 0, y(0) = y(π) = 0 (Sturm-Liouville). λ > 0: y = Asin(√λx)+Bcos(√λx). y(0) = B = 0. y(π) = Asin(√λπ) = 0: √λπ = nπ: λₙ = n²! Eigenfunctions: sin(nx)! Orthogonal: ∫₀^π sin(nx)sin(mx) = 0 (n≠m) ✓! (Fourier sine series!)