Understanding Laplace Transforms: Solving Problems and Applying Properties

In summary, the conversation revolved around two questions regarding Laplace transforms. The first question involved simplifying an expression using the linearity property and understanding the steps to go from one form to another. The second question was about the use of convolution and the definitions of sinh(t) and cosh(t) in terms of exponentials. The conversation also mentioned the importance of understanding the basics of Laplace transforms, specifically the LT of e^t and e^-t, and provided additional resources for further reading.
  • #1
LostEngKid
12
0
Hi all, 2 questions here

1) I've been doing some questions on laplace transforms and have been running into some trouble getting my answers into the same form as the answers given with the questions.
For example:

f(t) = 1 - e^(-t)
Using the linearity property i got 1/s - 1/(s-1) which is correct? But the answers give 1/s(s-1). Later in the lecture notes they show how to do the inverse laplace transform of the answer they gave and use partial fractions to get it into the form of the answer I got, i was just wondering what they have done to go from my answer to their answer, what steps am i missing?

2) f(t) = sinhtcosht, does this involve convolution or is that only when doing the inverse laplace transform where L(f)L(g) = f(t)*g(t)?
 
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  • #2
Its been a while since I've looked at Laplace transforms, but is the LT of e^-t not 1/(s+1) instead of 1/(s-1)?
 
  • #3
1) Try and simplify your expression using a common denominator of s and (s-1).

2) Do you know the definition of sinh(t) and cosh(t) in terms of exponentials?
 
  • #4
oops yeah i did mean 1/(s+1) danago

and Pere i wouldn't have a clue about sinh(t) and cosh(t) a exponentials, maybe i should have paid more attention in first year maths
 
  • #5
LostEngKid said:
oops yeah i did mean 1/(s+1) danago

and Pere i wouldn't have a clue about sinh(t) and cosh(t) a exponentials, maybe i should have paid more attention in first year maths


If you don't know about the definition of sinh(t) I don't see how you could calculate its LT..

[tex]
\begin{align*}
\sinh t =& \frac{e^t-e^{-t}}{2}\\
\cosh t =& \frac{e^t+e^{-t}}{2}
\end{align}
[/tex]

For some background reading you could have a look at http://en.wikipedia.org/wiki/Hyperbolic_trigonometric_function" [Broken].
 
Last edited by a moderator:
  • #6
There should be a table of Laplace transforms which you can refer to check LT of sinh and cosh. But in any case, Pere Callahan's post is sufficient for you to do it, so long as you know the LT of e^t and e^-t.
 

1. What is a Laplace Transform?

A Laplace Transform is a mathematical operation used to convert a function of time into a function of complex frequency. It is often used in engineering and science to solve differential equations and analyze systems.

2. How do I perform a Laplace Transform?

To perform a Laplace Transform, you need to take the integral of the function of time multiplied by the exponential function of negative time. This integral can be evaluated using tables or software, such as MATLAB or Wolfram Alpha.

3. What are the benefits of using Laplace Transforms?

Laplace Transforms have several benefits, including simplifying the process of solving differential equations, allowing for the use of algebraic operations on functions, and providing a better understanding of system behavior in the frequency domain.

4. Can Laplace Transforms be used for any type of function?

Yes, Laplace Transforms can be used for any continuous function. However, the function must be defined for all positive values of time.

5. Is there a way to invert a Laplace Transform?

Yes, it is possible to invert a Laplace Transform using a table or software. However, the inverse Laplace Transform may not always exist for every function, and it may be difficult to find the inverse in some cases.

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