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assume auxiliary equation boundary value problem calculate Chapter Consider the equation Consider the system constant coefficients defined derivatives difference equation eigenfunctions eigenvalues Elementary Systems equations with constant Euler's method Example formula Fourier series fundamental matrix solution gallons given Hence homogeneous equation homogeneous system initial conditions initial value problem integral interval Laplace transform Legendre polynomials Linear Second Order linear system linearly independent solutions nonzero obtain Order Difference Equations Order Differential Equations ordinary differential equations Partial Differential Equations particular solution polynomial power series method Power Series Solutions proof Prove radius of convergence satisfies Second Order Difference Second Order Differential Section Show sin2 sinh solution of Eq solution pairs Solutions of Differential solve Sturm-Liouville problem substitution Suppose system x Systems of Differential tank Theorem unique solution vanish variables vector Wronskian yields yp(x