The Laplace Transform

Evaluating the integral of e^(-st) f(t) from 0 to infinity for standard functions, recovering f from F by the inverse, and exploiting linearity of both.

The ODE Project, The Laplace Transform

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21 exercises

Exercise 1

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = \sin at$

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Exercise 2

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = \cos at$

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Exercise 3

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = e^{at} \sin bt$

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Exercise 4

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = \cosh at$

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Exercise 5

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = 2 \cos at - 3\sinh at$

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Exercise 6

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = t e^{at}$

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Exercise 7

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = t \sin at$

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Exercise 8

Find the Laplace transform for each of the functions in Finding Laplace transforms. Recall that $\cosh t = (e^t + e^{-t})/2$ and $\sinh t = (e^t - e^{-t})/2$ .

$f(t) = t^2 \cos at$

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Exercise 9

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{1}{s(s + 1)}$

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Exercise 10

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = s^{-3/2}$

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Exercise 11

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{1}{s + 7}$

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Exercise 12

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{2}{s^2 + 9}$

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Exercise 13

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{5s - 4}{2s^2 + s - 1}$

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Exercise 14

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{2s^2 - s + 4}{s^3 + 4s}$

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Exercise 15

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{1}{(s + 2)^3}$

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Exercise 16

Find the inverse Laplace transform for each of the functions in Finding inverse Laplace transforms. You will find partial fraction decomposition very useful.

$F(s) = \dfrac{1}{(s + 1)^2(s^2 - 9)}$

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Exercise 17

Prove this theorem: If $f$ is a piecewise continuous, exponentially bounded function defined $[0, \infty)$ , then the Laplace transform of $f$ , exists.

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Exercise 18

Let $f$ be a function whose Laplace transform ${\mathcal L}(f)(s)= F(s)$ exists for $s \gt a$ . Prove that for $s \gt a + c$ .

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Exercise 19

Prove this theorem: Suppose that $f$ and $g$ are piecewise continuous, exponentially bounded functions defined $[0, \infty)$ , with Laplace transforms of $F(s)$ and $G(s)$ , respectively. If $F(s) = G(s)$ , then $f(t) = g(t)$ .

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Exercise 20

If $f$ is a piecewise continuous, exponentially bounded function defined $[0, \infty)$ , prove that the Laplace transform of $f$ , exists.

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Exercise 21

Define the gamma function to be for $x \gt 0$ . The gamma function is very useful for expressing the Laplace transform of the function $t^\alpha$ .

Show that $\Gamma(1) = 1$ .
Prove that $\Gamma(x + 1) = x\Gamma(x)$ and deduce that $\Gamma(n+1) = n!$ for $n \in {\mathbb N}$ .
Show that the Laplace transform of $f(t) = t^\alpha$ is If $n$ is a positive integer, then ${\mathcal L}(t^n)(s) = n!/s^{n + 1}$ .

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