Having difficulty finding the inverse laplace transform

In summary, the conversation is about finding the inverse Laplace transform for a given function, and the difficulties the speaker is facing with rewriting the denominator and factoring it. The expert suggests rewriting the exponential term, using a familiar transform, and shifting the t-domain functions. However, the speaker is still struggling and cannot split the denominator into two linear factors. The expert then suggests completing the square instead, leading to a solution involving t-shifted, exponentially damped sines and cosines.
  • #1
Theelectricchild
260
0
Having difficulty finding the inverse laplace transform!

Hello everyone, I am really stuck on finding the inverse Laplace transform for this:

[tex]f(s)=\frac{5se^{-3s} - e^{-3s}}{s^{2}-4s+17}[/tex]

Heres my reasoning: I feel that I should rewrite the denominator in some kind of form such as (s-2)^2 + 13, and note the similarity with some of the problems I've been doing before, however its that 13 that is bothering me! Its not something you can take the squareroot of--- and also in addtion, I tried factoring out e^-3s on top and splitting this into two equations, but its this denominator that I absolutely despise.

Any help with finding the right method would be greatly appreciated thank you!
 
Last edited:
Physics news on Phys.org
  • #2
I've forgotten just about every transform formula I knew (and I'm not in the mood rederiving them), but you may write [tex]13=(\sqrt{13})^{2}[/tex]

if that helps..
 
  • #3
hmm i still am having difficulty---
 
  • #4
Well, what about rewriting:
[tex]e^{-3s}=e^{-6}e^{-3(s-2)}[/tex]

Then you would get one expression on the form:
[tex]k\frac{e^{-3w}}{w^{2}+a^{2}},k=-e^{-6},w=s-2,a=\sqrt{13}[/tex]

Is this a familiar transform in w?
 
  • #5
The exponentials are due to shifts in the t-domain:

L{f(t-T)}=e-sTF(s).

Just find the inverse transform of the rational functions of s, and then let the t-domain functions be delayed by the appropriate amount, given by the coefficient in the exponents of the exp functions.
 
  • #6
interesting let me see if i can get anywhere with that...
 
  • #7
Uh oh, you see, I cannot split the denominator into two linear factors like the book seems to be doing with most of the problems---- and because I cant, i don't know how to proceed like the examples do... I am sorry if I am a bit slow but we just started Laplace and this is a challenge problem id like to know to prepare.
 
Last edited:
  • #8
Well I cannot solve it but thanks for your help anyway...
 
  • #9
Theelectricchild said:
Uh oh, you see, I cannot split the denominator into two linear factors like the book seems to be doing with most of the problems---- and because I cant, i don't know how to proceed like the examples do... I am sorry if I am a bit slow but we just started Laplace and this is a challenge problem id like to know to prepare.

You could factor the denominator into 2 linear factors, but the roots of that polynomial are complex. It would be better instead to complete the square in the denominator. The solution will be t-shifted, exponentially damped sines and cosines.
 

1. What is the inverse Laplace transform?

The inverse Laplace transform is a mathematical operation that takes a function in the Laplace domain and converts it back to its original form in the time domain. It is essentially the reverse of the Laplace transform.

2. Why is it important to find the inverse Laplace transform?

The inverse Laplace transform is important because it allows us to solve differential equations in the time domain by converting them to algebraic equations in the Laplace domain. It is also useful in signal processing and control theory.

3. What makes finding the inverse Laplace transform difficult?

Finding the inverse Laplace transform can be difficult because it involves complex mathematical operations and requires knowledge of various techniques and formulas. It also requires a good understanding of the properties and behavior of Laplace transforms.

4. What are some common techniques for finding the inverse Laplace transform?

Some common techniques for finding the inverse Laplace transform include partial fraction decomposition, convolution, and the use of tables and formulas. It is important to have a good understanding of these techniques and when to use them.

5. How can I improve my ability to find the inverse Laplace transform?

Improving your ability to find the inverse Laplace transform requires practice and a solid understanding of the underlying concepts. It can also be helpful to work through examples and seek guidance from experts or online resources. Additionally, familiarizing yourself with commonly used techniques and formulas can make the process easier.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
4
Views
975
  • Engineering and Comp Sci Homework Help
Replies
3
Views
1K
  • Differential Equations
Replies
17
Views
789
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Differential Equations
Replies
1
Views
571
  • Introductory Physics Homework Help
Replies
5
Views
3K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
Back
Top