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What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will prove below is key to all that follows.
Let be a function such that (you'll need this assumption in the following). Use integration by parts to show the following about Laplace transforms of derivatives.
Hint: makes an excellent .
OK, so I've said so far that
I let my u = and let dv = , du = -s and v = but then I get confused about how to implement the definite integral form of integration by parts, and won't I have to use this integration by parts several times?
After getting the integration by parts finished, I know I will have to take the limit as b approaches infinity from 0 to b. Then I can maybe apply L'Hopital's rule to simplify and prove that ?
The Laplace transform of a function and its derivative are investigated. The solution is detailed and well presented.