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Section 3.1 Intro to Linear Systems

A homogeneous linear system with a constant matrix looks like

\begin{align*} x'(t) \amp = ax(t) + by(t)\\ y'(t) \amp = cx(t) + dy(t) \end{align*}

for constants \(a,b,c,d\text{.}\)

This is less of a handful to look at when we set \(\mathbf{x}(t) = \begin{bmatrix} x(t) \\ y(t) \end{bmatrix}\text{,}\) so that we have

\begin{align*} \mathbf{x}'(t) \amp = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix} \mathbf{x}(t). \end{align*}

Typically, a homogeneous linear system will look like \(\mathbf{x}' = A \mathbf{x}\) for some constant matrix \(A\)

More generally, a homogeneous linear system looks like \(\mathbf{x}' = P(t) \mathbf{x}\text{.}\) This chapter will mainly go over the case with \(P(t) = A\text{.}\)

Now, as one might guess, a nonhomogeneous linear system looks like

\begin{align*} \mathbf{x}'(t) \amp = A \mathbf{x}(t) + \mathbf{g}(t). \end{align*}

We'll go over how to solve the above linear system (with some conditions that usually apply).