Question

Section 8.1 Introduction to the Laplace Transforms

Problem 8.

Use the known Laplace transform L(1)=1/s and the result of Exercise 6 to show that
$L( {t}^{n} ) = \frac{n!}{ {s}^{n + 1} } , \: n = integer.$
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Answer 1

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