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faust18 [17]
1 year ago
15

Section 8.1 Introduction to the Laplace Transforms

Mathematics
1 answer:
sukhopar [10]1 year ago
7 0

Presumably you've proven exercise 6, that the Laplace transform of t^k f(t) is (-1)^k F^{(k)}(s).

Let F(s) = 1/s, whose inverse Laplace transform is f(t) = 1. Differentiate F with respect to s :

F'(s) = -\dfrac1{s^2}

By the claim from ex.6, this is the Laplace transform of t • f(t) = t.

Differentiate F again with respect to s :

F''(s) = \dfrac2{s^3}

and this is the Laplace transform of t² • f(t) = t². And so on.

We can prove the general claim by induction. Assume it's true for n = k, that t^k \leftrightarrow \frac{k!}{s^{k+1}}. Then using the result of ex.6, we have

F(s) = \dfrac{k!}{s^{k+1}} \implies F'(s) = -\dfrac{(k+1)!}{s^{k+2}} \leftrightarrow t^{k+1}

QED

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