Laplace Transformation help~

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plz help me with this question

Find the laplace transformation of this function

i(t)=(t)(e^t)(sinkt)

i really dont know how to do!

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The Laplace Transform is defined as:

$$Y(s) = \int_{0}^{\infty} e^{-st}y(t)dt$$

where y(t) is the function you wish to find the Laplacian of.

In this example, the integral would be:

$$\int_{0}^{\infty} te^{-st}e^tsin(kt)dt$$

...which is unbelievably ugly.

Have you learned about convolution yet? This is a pretty nasty problem, unless I'm missing something, which it seems probable that I am.

Last edited:
thz

You didnt miss anything, i can do up to this stage, but it contain 3 t in it, I dont really know how to solve it!

arildno
Homework Helper
Gold Member
Dearly Missed
1. Since the integral of the e^((1-s)t)*sin(kt) will "rotate" during integration by parts (i.e. you will gain back a multiple of what you began integrating), evaluating the integral of this function alone should pose no problems.
(Assuming s>1, that is)

2. You can now go back to the original problem, using integration by parts to eliminate the t-factor.

3. Alternatively, you might use the complex exponential as a simplifying measure.

Tom Mattson
Staff Emeritus
Gold Member
arildno said:
3. Alternatively, you might use the complex exponential as a simplifying measure.
That's what I would do, too. The beautiful thing about that is that, not only is it a lot easier to calculate, but it also gives you TWO Laplace transforms simultaneously.

mak_wilson, I would recommend that you take this suggestion. Make the replacement:

sin(kt)--->eikt

and take the imaginary part at the end.

solution

Here is a solution,
Max.

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thank You~~