Differential Equations and Linear Algebra, 4th Edition

A.2 Linear Systems

An important application of the global existence theorem just given is to the initial value problem

$$\frac{d\mathbf{\text{x}}}{dt}=\mathbf{\text{A}}(t)\mathbf{\text{x}}+\mathbf{\text{g}}(t),\mathbf{\text{x}}(a)=\mathbf{\text{b}}$$ (29)

for a linear system, where the $m\times m$ matrix-valued function A(t) and the vector-valued function g(t) are continuous on a (bounded or unbounded) open interval I containing the point $t=a.$ In order to apply Theorem 1 to the linear system in (29), we note first that the proof of Theorem 1 requires only that, for each closed and bounded subinterval J of I, there exists a Lipschitz constant k such that

$$|\mathbf{\text{f}}({\mathbf{\text{x}}}_1,t)-\mathbf{\text{f}}({\mathbf{\text{x}}}_2,t)|\leqq k|{\mathbf{\text{x}}}_1-{\mathbf{\text{x}}}_2|$$ (20)

for all t in J (and all ${\mathbf{\text{x}}}_1$ and ${\mathbf{\text{x}}}_2$). Thus we do not need a single Lipschitz constant for the entire open interval I.

In (29) we have $\mathbf{\text{f}}(\mathbf{\text{x}},t)=\mathbf{\text{A}}(t)\mathbf{\text{x}}+\mathbf{\text{g}},$ so

$$\mathbf{\text{f}}({\mathbf{\text{x}}}_1,t)-\mathbf{\text{f}}({\mathbf{\text{x}}}_2,t)=\mathbf{\text{A}}($$