Laplace Transforms 10: Examples with the Heaviside Function
(Video 10 of more to come) In the last video, we met the Heaviside function, also known as the unit step function, and discussed its utility in delaying the start of an external force. This video focuses on applying the Heaviside function to turn on a signal after a certain time. We examine two examples: a basic first-order differential equation and a more complex second-order one.
Error: at 9:14, I should write 3y(0), but luckily, y(0)=0, so it does not affect the final result.
Our first example involves the differential equation dy/dt = H(t - 2). The solution is a piece-wise function that remains constant at one until \( t = 2 \) and then linearly increases. For the second example, we solve a second-order differential equation y'' + 3y' + 2y = H(t - pi)sin(2(t - pi)). This equation indicates the onset of a sinusoidal forcing function at t = pi. We use the Laplace transform to solve this equation, incorporating initial conditions and applying identities for first and second derivatives. The solution involves partial fraction decomposition and the Laplace transform of delayed signals.
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