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17Calculus - Laplace Transforms

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Laplace Transforms

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Laplace Transforms involve a technique to change an expression into another form using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, they require only improper integration techniques to use. So you may run across them in first year calculus.

The Laplace Transform

\(\displaystyle{ \mathcal{L}\{ f(t) \} = F(s) = \int_{0}^{\infty}{e^{-st}f(t)~dt} }\)

In the integral, \(s\) is considered a constant and \(s > 0\)

After integration, we have a function \(F(s)\), which is a function of \(s\) (now considered a variable)

Overview

This takes some thinking and getting used to. So let's start off by watching some videos. There is some duplication coming up but you really need to have that repetition in order to get your head around this concept.

This first video is a quick overview with some calculations to get you started. Do not worry too much about understanding everything here since we will revisit these concepts very soon. Just get the big picture.

Dr Chris Tisdell - Intro to Laplace transform and how to calculate them [8min-48secs]

video by Dr Chris Tisdell

Now that you have the big picture, let's start working on understanding the details of Laplace Transforms. This is a great video clip explaining this notation and the improper integral definition. It is relatively short but is packed with good information including explaining notation and calculating the integral.

Dr Chris Tisdell - Introduction to Laplace transforms (part 1) [5min-35secs]

video by Dr Chris Tisdell

Okay, as explained in that video, the Laplace Transform can only be used when the improper integral converges. So the next question is, what kind of functions can the Laplace Transform be applied to?

Piecewise Continuous Functions

This video clip explains one very important type of function, a piecewise continuous function, where the Laplace Transform really helps us with.

Dr Chris Tisdell - Introduction to Laplace transforms (part 2) [2min-56secs]

video by Dr Chris Tisdell

Now, from that last video, you can tell from his graph that a piecewise continuous function is NOT a piecewise function that is continuous. It is a piecewise function that has sections that are continuous, like the plot on the right. Applying the Laplace Transform to these kind of functions is where the Laplace Transform really shines.

Also notice in the last video, that he calculated the Laplace Transform of \(\cos(x)\) and \(\sin(x)\) without using the improper integral directly, which would require integration by parts. That is one of the powerful things about Laplace Transforms.

Table of Laplace Transforms

Fortunately, you are probably not going to be asked to use the improper integral at the top of the page to calculate most Laplace Transforms. You will use a Laplace Transform that you know and derive what you are looking for. In order to do that, you will usually be given a table of Laplace Transforms. These will also be used to get the inverse Laplace Transform (since no direct method exists that we can understand at this level of mathematics).

This video clip finishes the video shown previously and demonstrates how to use tables of Laplace Transforms.

Dr Chris Tisdell - Introduction to Laplace transforms (part 3) [13min-47secs]

video by Dr Chris Tisdell

Okay, now that you see the importance of tables for Laplace Transforms, you are probably expecting to see a table, right? Well, here is where you can find one. We highly recommend this book. It is free for you to download.
Engineering Math Workbook by Dr Chris Tisdell

Here is a list of some Laplace Transforms. We suggest you build your own list based on these, the workbook above and the pages listed in the related topics panel.

Laplace Transforms Table

Laplace Transforms

\( f(t) \)

\(\displaystyle{ F(s) }\)

Practice

Basic Functions

\( t^n, ~ n = 1, 2, 3, \ldots \)

\(\displaystyle{ \frac{n!}{s^{n+1}} }\)

650

\( e^{at} \)

\(\displaystyle{ \frac{1}{s-a} }\)

3604

\( \sin(\alpha t) \)

\(\displaystyle{ \frac{\alpha}{s^2 + \alpha^2} }\)

652

\( \cos(at) \)

\(\displaystyle{ \frac{s}{s^2 + a^2} }\)

\( \sinh(at) \)

\(\displaystyle{ \frac{a}{s^2 - a^2} }\)

\( \cosh(at) \)

\(\displaystyle{ \frac{s}{s^2 - a^2} }\)

3607, 655

Special Functions

\( \delta(t) \) unit impulse

\( 1 \)

\( \delta(t-\tau) \) shifted unit impulse

\( e^{-\tau s} \)

\( u(t) \) unit step

\(\displaystyle{ \frac{1}{s} }\)

\( u(t-\tau) \) shifted unit step

\(\displaystyle{ \frac{1}{s} e^{-\tau s} }\)

Combined Functions

\( e^{at}\sin(\alpha t) \)

\(\displaystyle{ \frac{\alpha}{(s-a)^2 + \alpha^2} }\)

\( f(t)u(t-a) \)

\(\displaystyle{ e^{-sa} \mathcal{L}\{ f(t+a) \} }\)

\( t^n e^{at}, ~ n = 1, 2, 3, \ldots \)

\(\displaystyle{ \frac{n!}{(s-a)^{n+1}} \} }\)

Derivatives and Integrals

\( f'(t) \)

\( sF(s) - f(0) \)

\( f''(t) \)

\( s^2F(s) - sf(0) - f'(0) \)

\( \int_0^t{ f(v)~dv }\)

\(\displaystyle{ \frac{F(s)}{s} }\)

\(\displaystyle{ f^{(n)}(t) }\)

\(\displaystyle{ s^nF(s) - s^{n-1}f(0) - }\) \(\displaystyle{ s^{n-2}f'(0) - . . . - f^{(n-1)}(0) }\)

Examples

Okay, let's look at a couple of examples. Try them on your own first. The tools you need are
1. the definition of the Laplace Transform at the top of the page
2. improper integral techniques (Make sure you use correct notation including the limit.)
3. integration by substitution
4. integration by parts
You should know all these by now, so you know how to do these calculations. The solutions are shown in video clips.

Example 1

Calculate the Laplace Transform of \(\displaystyle{ f(t) = e^{-at} }\).

This video clip shows the step-by-step solution. It also demonstrates the incredible power of the Laplace Transform. It is a little longer than the previous two video clips but make sure and watch the entire clip. Your time will be well spent.

Dr Chris Tisdell - Introduction to Laplace transforms (part 4) [21min-49secs]

video by Dr Chris Tisdell

Example 2

Calculate the Laplace Transform of \(tf(t)\).

Before you go to the practice problems, watch this video showing how to calculate the Laplace transform of \(tf(t)\). Yes, this probably appears your table, but it will help you get a better feel for Laplace transforms if you work through this video with the instructor. (If you feel adventurous, try working it on your own before watching the video. There is nothing new here, just an improper integral that uses integration by parts.)

Dr Chris Tisdell - Laplace Transform of tf(t) [7min-24secs]

video by Dr Chris Tisdell

Laplace Transform of a Periodic Function

A useful concept of the Laplace Transform is how it works with periodic functions. Periodic functions are, of course, functions that repeat in a periodic fashion. You've seen periodic functions in trigonometry. Sine and cosine are both periodic with period \(T=2\pi\).

The equation to find the Laplace Transform of a periodic function \(f(t)\) with period \(T\) is \(\displaystyle{ \mathcal{L}\{ f(t) \} = \frac{1}{1-e^{-sT}} \int_{0}^{T}{ e^{-st}f(t)~dt } }\)

Here is a great video clip showing the derivation of the last equation. Don't skip this video. It has important information that will help you when working with Laplace Transforms.

Dr Chris Tisdell - Laplace transform: square wave [8min-51secs]

video by Dr Chris Tisdell

Before you go on, work these practice problems. After that you need to learn how to go the other direction, i.e. when given the inverse laplace transform, how to find the original function.

Practice

Unless otherwise instructed, calculate the Laplace Transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

Basic

\( f(t) = t \)

Problem Statement

For \( f(t) = t \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

Solution

650 video

video by PatrickJMT

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\( y(t) = e^{at} \)

Problem Statement

\( y(t) = e^{at} \)

Solution

3604 video

video by blackpenredpen

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Calculate \( \mathcal{L}\{ \sinh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).

Problem Statement

Calculate \( \mathcal{L}\{ \sinh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).

Solution

3607 video

video by blackpenredpen

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Calculate \( \mathcal{L}\{ \cosh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).

Problem Statement

Calculate \( \mathcal{L}\{ \cosh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).

Solution

3608 video

video by blackpenredpen

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For a constant a > 0, if \( \mathcal{L}\{ f(t) \} = F(s) \), show that \(\displaystyle{ \mathcal{L}\{ f(at) \} = \frac{1}{a}F(s/a) }\)

Problem Statement

For a constant a > 0, if \( \mathcal{L}\{ f(t) \} = F(s) \), show that \(\displaystyle{ \mathcal{L}\{ f(at) \} = \frac{1}{a}F(s/a) }\)

Hint

Start by using the substitution \(u=at\) in the integral definition of the Laplace Transform.

Problem Statement

For a constant a > 0, if \( \mathcal{L}\{ f(t) \} = F(s) \), show that \(\displaystyle{ \mathcal{L}\{ f(at) \} = \frac{1}{a}F(s/a) }\)

Hint

Start by using the substitution \(u=at\) in the integral definition of the Laplace Transform.

Solution

3615 video

video by blackpenredpen

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\( \mathcal{L}\{ e^{at}f(t) \} \)

Problem Statement

\( \mathcal{L}\{ e^{at}f(t) \} \)

Solution

3616 video

video by blackpenredpen

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\( \mathcal{L}\{ e^{3t}\cos(2t) \} \)

Problem Statement

\( \mathcal{L}\{ e^{3t}\cos(2t) \} \)

Solution

3618 video

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\( \mathcal{L}\{ t f(t) \} \)

Problem Statement

\( \mathcal{L}\{ t f(t) \} \)

Solution

3619 video

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Evaluate \( \mathcal{L}\{ t \sin(bt) \} \) using a table.

Problem Statement

Evaluate \( \mathcal{L}\{ t \sin(bt) \} \) using a table.

Solution

3620 video

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Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using a table.

Problem Statement

Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using a table.

Solution

3617 video

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Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using \( \mathcal{L}\{ t^n f(t) \} = (-1)^n F^{(n)}(s) \)

Problem Statement

Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using \( \mathcal{L}\{ t^n f(t) \} = (-1)^n F^{(n)}(s) \)

Hint

\( F^{(n)}(s) \) is the nth-derivative of \( F(s) \).

Problem Statement

Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using \( \mathcal{L}\{ t^n f(t) \} = (-1)^n F^{(n)}(s) \)

Hint

\( F^{(n)}(s) \) is the nth-derivative of \( F(s) \).

Solution

3621 video

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\( f(t) = e^{3t} + \cos(6t) - e^{3t}\cos(6t) \); use a table

Problem Statement

For \( f(t) = e^{3t} + \cos(6t) - e^{3t}\cos(6t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a table.

Solution

653 video

video by Krista King Math

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Intermediate

Find \( \mathcal{L}\{ \sin(bt) \} \) and \( \mathcal{L}\{ \cos(bt) \} \) using Euler's Formula.

Problem Statement

Find \( \mathcal{L}\{ \sin(bt) \} \) and \( \mathcal{L}\{ \cos(bt) \} \) using Euler's Formula.

Solution

3606 video

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\( f(t) = \sin(3t) \)

Problem Statement

For \( f(t) = \sin(3t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

Solution

652 video

video by Krista King Math

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Advanced

Calculate \( \mathcal{L}\{ t^n \} \) using power series.

Problem Statement

Calculate \( \mathcal{L}\{ t^n \} \) using power series.

Hint

Use \( \mathcal{L}\{ e^{at} \} \) from a previous problem and then find it's power series.

Problem Statement

Calculate \( \mathcal{L}\{ t^n \} \) using power series.

Hint

Use \( \mathcal{L}\{ e^{at} \} \) from a previous problem and then find it's power series.

Solution

3605 video

video by blackpenredpen

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\( f(t) = t \cosh(3t) \)

Problem Statement

For \( f(t) = t \cosh(3t) \), find the Laplace transform using the integral definition.

Solution

655 video

video by Krista King Math

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\( \mathcal{L}\{ f(t)/t \} \)

Problem Statement

\( \mathcal{L}\{ f(t)/t \} \)

Solution

3622 video

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Evaluate \( \mathcal{L}\{ \sin(t)/t \} \) using a table.

Problem Statement

Evaluate \( \mathcal{L}\{ \sin(t)/t \} \) using a table.

Solution

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You CAN Ace Calculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Practice Instructions

Unless otherwise instructed, calculate the Laplace Transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

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