Linear Systems of Differential Equations with Forcing - Convolution and the Dirac Delta Function

Explore the derivation of the general solution to matrix systems of linear differential equations with forcing through convolution integrals in this 41-minute video lecture. Begin with a breakdown of simple sub-problems, including the system's response to an impulsive delta-function input. Discover the Dirac delta function and learn how a generic input forcing function can be viewed as a sequence of infinitesimal delta functions. Progress through various cases, including initial condition responses, impulse responses, and the combination of both. Culminate with the development of the convolution integral for arbitrary forcing, a fundamental concept in control theory and linear systems analysis. Gain insights into this cornerstone approach through clear explanations and step-by-step derivations.

Overview
Case 1: Initial condition response with no forcing
The Dirac delta function
Case 2: Impulse response for delta function input
Case 3: Impulse response with an initial condition
Convolution integral for arbitrary forcing ut