Laplace Transform of f(t) = t sin(2t)

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ODEs: Find the Laplace Transform of f(t) = t sin(2t). We use the derivative formula for the Laplace Transform: L(y') = sL(y) - y(0) instead of the definition. This method also finds the Laplace transform of g(t) = t cos(2t). We verify the answer by finding the IVP with f(t) as the unique solution.

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@SuperEricliao You're welcome, and thanks for the comment! I set this sequence up to bring in different features of LT with each video, and this turned out to be a nice application of the derivative rule. - Bob

MathDoctorBob

In the formula, we use y' in the derivative in L on the left, but not for y functions of the right. So y(t) = tcos(2t) goes to 0. - Bob

MathDoctorBob

This follows from the rule for exp(ax) cos(bx). Write sinh(3t) as (exp(3t) - exp(-3t))/2 and cos^2(t) as (1+cos(2t))/2. Multiply and apply LT to each term in the sum. Hope that helps.

MathDoctorBob

Correct me please if I'm wrong, but isn't there another way of doing it?
L[t^n.f(t)] = (-1)^n. (d^n/ds^n)F(s)?

amadeuszjacobs

Note that
f(t)=-(1/4)cos(4t) using the half angles formulas twice.

MathDoctorBob

@estossala1 You're welcome, and thanks for the kind words! - Bob

MathDoctorBob

Your first step is to IBP the t out of the problem. That leaves you with an integral of e^ax sin(bx), which no one loves. The derivative formula removes the bookkeeping.

MathDoctorBob

Thanks for the video Bob. This is actually what I was looking for to solve laplace problems using direct integration.

Davidgilmour

@RE90 You're welcome! Glad to be of help. - Bob

MathDoctorBob

Thanks! It's a nice way to motivate the formula for derivatives. - Bob

MathDoctorBob

Check out the video Laplace Transform of f(t) = t^2 e^{2t} cos(3t). Near the end, Warmup 2, I show how to do t^2cos(3t). Once you have a few rules, it becomes manipulations.

MathDoctorBob