4.3 The inhomogeneous system
Theorem 4.3.1 (Variation of parameters). The solution of \(x' = A(t)x + g(t)\), \(x(t_0) = x_0\), is \[ x(t) = \Phi (t, t_0)x_0 + \int _{t_0}^t\Phi (t, s)g(s)\,ds . \]
Proof. Seek a solution of the form \(x = \Phi (t)c(t)\), letting the constants of the homogeneous solution vary. Then \(x' = A\Phi c + \Phi c'\), which equals \(A\Phi c + g\) if and only if \(\Phi c' = g\), that is \(c' = \Phi (t)^{-1}g(t)\). Integrating with \(c(t_0) = \Phi (t_0)^{-1}x_0\), \[ x(t) = \Phi (t)\Phi (t_0)^{-1}x_0 + \int _{t_0}^t\Phi (t)\Phi (s)^{-1}g(s)\,ds , \] which is the formula. Since the problem has a unique solution, this is it. □
The formula is Duhamel’s principle: the response to the forcing \(g\) is the superposition of the responses to impulses \(g(s)\,ds\) applied at each earlier time \(s\), each propagated forward by the homogeneous flow \(\Phi (t, s)\). The general solution is a particular solution plus the general homogeneous solution, the pattern of Theorem 1.2.1 in \(n\) dimensions.
For a second-order scalar equation \(y'' + p(t)y' + q(t)y = g(t)\) with independent homogeneous solutions \(y_1\), \(y_2\) and Wronskian \(W\), the formula reduces to \[ y_p(t) = -y_1(t)\int \frac {y_2(t)g(t)}{W(t)}\,dt + y_2(t)\int \frac {y_1(t)g(t)}{W(t)}\,dt , \] which is the classical method of variation of parameters (Problem 4.7.3). Written as a single integral, \(y_p(t) = \int _{t_0}^tG(t, s)g(s)\,ds\) with \(G(t, s) = \bigl (y_1(s)y_2(t) - y_1(t)y_2(s)\bigr )/W(s)\), it introduces the Green’s function of the initial value problem, studied with its boundary value counterpart later in the course.
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