Struggling with Laplace transforms

In summary, the conversation discusses the concept of Laplace transforms and their application in solving problems in the frequency domain. The conversation also includes a discussion on a specific transform and its corresponding formula, as well as the importance of understanding derivatives in solving Laplace transform problems. There is also a mention of the unit step function and its role in the Laplace transform.
  • #1
Plecto
8
0
Hi. We are learning about Laplace transforms at uni and I must say that this is a real pain. I have one questions concerning the concept of Laplace transforms, and also a question concerning a specific transform. The task is to make a Laplace transform of: t*sin(2t). I could do an integration by parts and solve it using the standard definition of the Laplace transform, but I don't think that is the idea. The question says "L[f(t)]=F(s). L[t*f(t)]=-dF(s)/ds". I can't find this anywhere in the book other than above the assignment I'm asked to do so there's no explanation of it, I have no idea of what it means :( I was thinking that it might have to do with that L(f')=s*L(f)-f(0), but to use that, I would have to know the Laplace transform of f'(t), but I don't :( Is there anyone that could give me some help?

I'm also struggling to see what the frequency or s-domain actually tells me. Our lecturer gave an example where three sine waves were on top of each other and that it would be difficult to see exactly how many and at what frequencies they were. By doing a Laplace transform we could see the frequency along the s-axis and their amplitude along the y-axis, but what about doing the Laplace transform of a constant? A constant doesn't have a frequency, neither does a function like e^t. The Laplace transform of e^t is 1/s, what kind of information will that function give me? I can set s=2 which will give y=0.5, does it say that the amplitude is 0.5 when the frequency is 2hz? That makes no sense at all :(
 
Mathematics news on Phys.org
  • #2
A good tutorial will help you get started: http://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx

In your particular problem they have given you a special "product formula", for L[t*f(t)], where f(t) is any function such that L[f(t)] = F(s) ... so it is completely general.

The L[t*f(t)] = -d/ds[F(s)] ... you just take the derivative of F(s) times minus one.

In general, the Laplace transform is a member of a family of "integral transforms"; they create a 1-1 mapping from the domain where you started (time domain here) to a new domain (frequency domain) - and they are used because the problems are easier to solve in the new domain; then you transform back to the original domain.

In many cases there is a physical interpretation of the transform, and when you become familiar with it you can analyze problems in the s-domain without needing to go back and forth. The Fourier transform is a special case of the Laplace transform, and with just a bit of exposure people are "happy" to think in the frequency domain. The same happens in the s-domain, but perhaps it takes a bit longer!

This set of notes almost derives the formula which you have; you can easily finish the derivation:
http://www.stanford.edu/~boyd/ee102/laplace.pdf
 
  • #3
Alright, thanks for the response :) I had math at high school (equivalent) in 2008, I then took first year uni math in 2010 (barely made it), and now I'm taking second year uni math so there's a lot of holes in my knowledge :( I'm struggling to remember simple integration, differentiation etc.

So I take the Laplace transform of sin(2t) which gives me 2/(s^2+4), I then take the derivative of that and multiply it by -1 which gives me 4s/(s^2+4)^2 which is the right answer :) I don't get the right answer when trying to do the laplace transform of t*cos(3t) though. The Laplace transform of cos(3t) is s/(s^2+9), taking the derivative and multiplying by -1 gives me (-9-s^2)/(s^2+9)^2. The correct answer should be (s^2-9)/(s^2+9)^2 so I'm not sure what I'm doing wrong :(

I'll take a further look at the tutorial you posted. I've also watched a bunch of videos on youtube, but those video's doesn't seem to address everything.

The next question states that L[f(t)]/t=∫(from s to ∞) F(u). I think it's tiresome that these statements isn't mentioned anywhere else, no explanation or examples at all :( I don't know what F(u) is, I haven't worked with 'u' as a variable anywhere :( What function is F(u) exactly?
 
Last edited:
  • #4
Plecto said:
The Laplace transform of cos(3t) is s/(s^2+9), taking the derivative and multiplying by -1 gives me (-9-s^2)/(s^2+9)^2. The correct answer should be (s^2-9)/(s^2+9)^2 so I'm not sure what I'm doing wrong :(

The derivative should give 1/(s^2+9) - s(2s)/(s^2+9) = (9-s^2)/(s^2+9) ... which then gives the correct answer when multiplied by -1 for the t factor. So double-check your derivative to see where it went wrong.

Plecto said:
The next question states that L[f(t)]/t=∫(from s to ∞) F(u). I think it's tiresome that these statements isn't mentioned anywhere else, no explanation or examples at all :( I don't know what F(u) is, I haven't worked with 'u' as a variable anywhere :( What function is F(u) exactly?

u(t) is the unit step function; =0 for t<0, =1 for t>0. It is the function used to model a switch in a circuit. But I would need more context to be sure.
 
  • #5
Hi there,

Thank you for reaching out and sharing your struggles with Laplace transforms. It can definitely be a challenging concept to grasp, but with some practice and guidance, I'm sure you will be able to understand it better.

First, let's address your question about the specific transform of t*sin(2t). You are correct that using integration by parts and the standard definition of the Laplace transform is one way to solve this. However, there is another way to approach it using the property that you mentioned, L[t*f(t)]=-dF(s)/ds. This property is known as the differentiation property of Laplace transforms and it allows us to find the transform of a function that is multiplied by t. In this case, we can rewrite t*sin(2t) as t*(2/s^2+4^2), where the first term is the Laplace transform of sin(2t) and the second term is the Laplace transform of t. Using the differentiation property, we can then find the transform of t*sin(2t) as -dF(s)/ds of the transform of sin(2t), which is 4/s^2+4^2. I hope this helps clarify your doubts about this specific transform.

As for your question about the frequency and s-domain, it is true that a constant or a function like e^t does not have a specific frequency. However, the Laplace transform allows us to transform a function from the time domain to the frequency domain, which can be useful in analyzing systems and signals. The Laplace transform of a constant, as you mentioned, is 1/s, which tells us that the amplitude of the constant decreases as the frequency increases. Similarly, the Laplace transform of e^t tells us that the amplitude of the function decreases as the frequency increases. Setting s=2 and getting y=0.5 does not mean that the amplitude is 0.5 when the frequency is 2Hz. Instead, it means that the function e^t has an amplitude of 0.5 at a frequency of 2Hz. This information can be useful in analyzing systems that have exponential functions as inputs.

I hope this helps clarify some of your doubts about Laplace transforms. It's important to keep practicing and seeking help whenever needed. Don't get discouraged, with time and effort, you will be able to understand this concept better. Best of luck!
 

1. What are Laplace transforms and why do we use them?

Laplace transforms are mathematical tools used to solve differential equations. They transform a function of time into a function of frequency, making it easier to solve certain types of equations. They are commonly used in engineering and physics to solve problems involving systems that change over time.

2. How do I know when to use Laplace transforms?

Laplace transforms are typically used when solving differential equations with initial conditions. They are particularly useful for solving equations with non-constant coefficients, as they can transform the equation into a simpler form that is easier to solve.

3. What is the process for solving a problem using Laplace transforms?

The first step in solving a problem using Laplace transforms is to take the Laplace transform of both sides of the equation. This will transform the equation into an algebraic equation that can be solved for the transform of the desired function. Then, the inverse Laplace transform is taken to get the solution in terms of the original function. Finally, the solution is checked to ensure it satisfies the initial conditions given in the problem.

4. Are there any limitations or drawbacks to using Laplace transforms?

One limitation of Laplace transforms is that they can only be applied to linear differential equations. They also may not be suitable for problems with very complex initial conditions or for nonlinear systems. Additionally, some problems may require a large number of steps to solve using Laplace transforms, making it a time-consuming process.

5. How can I practice and improve my skills in using Laplace transforms?

The best way to improve your skills in using Laplace transforms is to practice solving problems. There are many resources available, such as textbooks, online tutorials, and practice problems, that can help you become more familiar with the process. It is also helpful to understand the underlying concepts and theory behind Laplace transforms, as this will make it easier to apply them to different types of problems.

Similar threads

  • General Math
Replies
3
Views
912
Replies
1
Views
737
  • General Math
Replies
1
Views
2K
Replies
1
Views
9K
  • Electrical Engineering
Replies
3
Views
938
Replies
0
Views
9K
Replies
1
Views
9K
Replies
2
Views
9K
Replies
1
Views
9K
  • Differential Equations
Replies
17
Views
806
Back
Top