Understanding the concept of laplace transformations

[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

 

[Corneo]

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

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

 

[M98Ranger]

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!