Laplace Transforms 12: Dirac Delta Examples

(Video 12 of more to come) In the last video, we introduced the Dirac delta as a model for a unit impulse at a given time t = a. We determined that the Laplace transform of this generalized function centered at a is e^(-sa). In this video, we will use the Laplace Transform to compute solutions to two differential equations where the right-hand side includes a Dirac delta function. These solutions match our expectations, showing the behavior of the system before and after the impulse. Our first example is a first-order differential equation: y' + 6y = delta(t - 1) with the initial condition y(0) = 3. We expect the solution to exhibit exponential decay until t = 1, at which point it is struck by an impulse, causing an immediate reaction. After the impulse, the solution resumes exponential decay. Next, we solve a second-order differential equation: y'' + 4y' + 13y = delta(t - 5) with initial conditions y(0) = 1 and y'(0) = 1. We expect the solution to exhibit underdamped harmonic oscillation with decreasing amplitude until t = 5, when it is struck by an impulse, causing a reaction, followed by continued oscillation with decreasing amplitude. Part 11: https://youtu.be/iSKkJcXIy7I #mathematics #math #laplacetransformation #laplacetransform #differentialequations #mathtutorial #engineeringmathematics #ordinarydifferentialequations #HeavisideFunction #UnitStepFunction #SignalProcessing #HarmonicOscillator