# Understanding the concept of laplace transformations

• slug
In summary, Laplace transformations involve transforming a function in terms of t into s and then taking the inverse transform to solve for the function. This is done by looking up the form of the function in a table and using the inverse transform to find the solution.
slug
I'm having trouble understanding the concept of laplace transformations.

my book states that it is comparing how much a function y(t) is like a standard function. what exactly does the answer mean such as y(s)=1/(s-2)
is this the differnence between the functions depending on the value of s

and further more how does taking the integral from zero to infinity of two functions tell me how much alike they are

stupid vague books i hate when they pull equations out of the air and don't explain where they come from and what they mean

The idea of Laplace Transformations is to transform a function in terms of t into s. Do some algebraic work and then take the inverse Laplace transform for the solution.

So given that you have a function in terms of s, that needs to be inversed transformed. You like it up in a table to see that your $Y(s) = \frac {1}{s-2}$ is in the form

$$F(s) = \frac {1}{s-a}$$ and its inverse transform is

$$\mathcal{L}^{-1} \{F(s)\} = e^{at}$$

So for you

$$\mathcal{L}^{-1} \{F(s)\} = e^{2t}$$

I completely understand your frustration with trying to understand the concept of Laplace transformations. It can be a difficult concept to grasp, but with some further explanation, I hope it will become clearer for you.

Firstly, the Laplace transformation is a mathematical tool that helps us solve differential equations. It involves converting a function of time (y(t)) into a function of a complex variable (y(s)).

Now, let's break down the equation y(s) = 1/(s-2). The variable s is known as the complex frequency and it represents the rate of change of the function y(t). In this case, the function y(t) is being compared to a standard function, which is 1/(s-2). This means that y(t) is similar to a function that has a pole at s=2. The pole is a point on the complex plane where the function becomes infinite. So, in simpler terms, the equation is telling us that the function y(t) behaves like 1/(s-2) at the point s=2.

Now, when we take the integral from zero to infinity of two functions, we are essentially comparing the areas under the curves of those functions. In the context of Laplace transformations, this helps us determine how similar or different two functions are. If the areas under the curves are the same, then the two functions are considered to be similar.

I understand your frustration with textbooks that seem to pull equations out of thin air without proper explanation. I would suggest seeking out additional resources or asking your professor for further clarification. With some practice and a deeper understanding of the concept, I am confident that you will be able to master Laplace transformations. Keep persevering and don't be afraid to ask for help. Best of luck!

## 1. What are Laplace transformations?

Laplace transformations are mathematical tools used to convert a function of time into a function of complex frequency. They are used to simplify the analysis of linear systems in the frequency domain.

## 2. Why are Laplace transformations useful?

Laplace transformations are useful because they allow us to solve differential equations and systems of equations more easily. They also provide a way to analyze the behavior of a system in the frequency domain, which can give insight into its stability and response to different inputs.

## 3. What is the difference between Laplace transformations and Fourier transformations?

The main difference between Laplace transformations and Fourier transformations is that Laplace transformations take into account the initial conditions of a system, while Fourier transformations do not. This makes Laplace transformations more useful for analyzing systems with non-zero initial conditions.

## 4. How do you perform a Laplace transformation?

To perform a Laplace transformation, you first need to find the Laplace transform of the function using a table or by using integration techniques. Then, you can use properties of Laplace transformations to simplify the transformed function. Finally, you can use an inverse Laplace transformation to find the original function.

## 5. What are some applications of Laplace transformations?

Laplace transformations are commonly used in engineering and physics to analyze and model systems in the frequency domain. They are also used in control theory, signal processing, and circuit analysis. Additionally, Laplace transformations have applications in probability and statistics, such as in the Laplace distribution.

• Differential Equations
Replies
17
Views
935
• Differential Equations
Replies
2
Views
2K
• Differential Equations
Replies
5
Views
4K
• Calculus and Beyond Homework Help
Replies
2
Views
943
• Engineering and Comp Sci Homework Help
Replies
4
Views
1K
• Differential Equations
Replies
1
Views
1K
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
1
Views
3K
• Differential Equations
Replies
1
Views
1K
• Differential Equations
Replies
2
Views
2K