6 Higher-Order Linear Equations and the Laplace Transform

Many models are stated directly as a single equation of order \(n\), such as Newton’s law for a mass on a spring, the current in a circuit, or the deflection of a beam. By Theorem 1.1.3 they are systems, and the theory of the previous two sections applies, but working with the scalar equation directly is often clearer and leads to the methods engineers use: characteristic polynomials, undetermined coefficients, impulse responses and the Laplace transform. This section develops them for the equation \[ L[y] = y^{(n)} + a_{n-1}y^{(n-1)} + \dots + a_1y' + a_0y = g(t) \] with constant real coefficients. Write \(D = d/dt\) and \(p(\lambda ) = \lambda ^n + a_{n-1}\lambda ^{n-1} + \dots + a_0\), the characteristic polynomial, so that \(L = p(D)\).

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