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: • Laplace Transforms 11: The Dirac Delt...
#mathematics #math #laplacetransformation #laplacetransform #differentialequations #mathtutorial #engineeringmathematics #ordinarydifferentialequations #HeavisideFunction #UnitStepFunction #SignalProcessing #HarmonicOscillator
Watch video Laplace Transforms 12: Dirac Delta Examples online without registration, duration hours minute second in high quality. This video was added by user Dr. Bevin Maultsby 20 May 2024, don't forget to share it with your friends and acquaintances, it has been viewed on our site 151 once and liked it 10 people.