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addition and scalar assertion belongs boundary conditions boundary-value problem closed interval compute continuous functions deduce defined definition derivative determined eigenvalues equation Ly equation of order equivalence relation Euclidean space Example exists exponential order fact FIGURE Find the Fourier finite dimensional finite number formula Fourier series Fourier series expansion given Green's function Hence Hint identity implies infinite initial conditions initial-value problem inner product integral inverse Laplace transform Legendre polynomials Lemma linear combination linear differential equation linear differential operator linear transformation linearly independent matrix non-negative nontrivial solution nonzero null space obtain one-to-one orthonormal set pair particular solution piecewise continuous piecewise continuous functions proof Prove real vector space respect result satisfies scalar multiplication Section sequence sin2 solution space solve spanned subset subspace suppose Theorem tion unique vectors in R2 Wronskian x-axis zero vector