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Section 3.1 Intro to Linear Systems

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A homogeneous linear system with a constant matrix looks like

x′(t)=ax(t)+by(t)y′(t)=cx(t)+dy(t)

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for constants a,b,c,d.

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This is less of a handful to look at when we set x(t)=[x(t)y(t)], so that we have

x′(t)=[abcd]x(t).

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Typically, a homogeneous linear system will look like x′=Ax 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 \(P(t) = A\text{.}\)

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

x′(t)=Ax(t)+g(t).

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