49 Motivation

Write the DE in the standard form of a first order linear ODE

$$\frac{dX}{dx}-AX=U$$

so the integrating factor is $\exp (-tA)$, and

$$\exp (-tA)\frac{dX}{dt}-\exp (-tA)AX=\exp (-tA)U$$

$$\frac{d}{dt}\left(\exp (-tA)X\right)=\exp (-tA)U,$$

by frame 31, so we integrate to get

$${\left[\exp (-tA)X\right]}_0^w={\int }_0^w\exp (-tA)U(t)𝑑t$$

$$\exp (-wA)X(w)-\exp (0)X(0)={\int }_0^w\exp (-tA)U(t)𝑑w$$