4 Linear Systems
Linear systems are the one large class of differential equations with a complete general theory. They describe small oscillations, electrical networks, compartment models in pharmacology, and every linearisation of a non-linear system about a solution, which is how they enter the stability theory later in the course. This section treats the system \[ x' = A(t)x + g(t), \qquad x \in \mathbb R^n, \] with \(A\) and \(g\) continuous on an open interval \(I\). By Theorem 3.2.1, every initial value problem has a unique solution on the whole of \(I\), so questions of existence are settled; what remains is the structure of the solutions. The next section specialises to constant \(A\), where the solutions can be written down.
4.2 Liouville’s formula
4.3 The inhomogeneous system
4.4 The adjoint system
4.5 When is the fundamental matrix an exponential?
4.6 Linear systems in software
4.7 Practice Problems
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