Finding the Laplace Transformation of a Piecewise Function

In summary, the Laplace transformation of the given function is equal to the second derivative of the integral of e^-(a+s)t from 0 to infinity. To evaluate this, one can use integration by parts.
  • #1
Northbysouth
249
2

Homework Statement


Obtain the Laplace transformation of the function defined by

f(t) = 0 t<0

= t2e-at t>=0

Homework Equations


The Attempt at a Solution



I'm a little unsure of what I'm doing here, so bear with me.

L {t2e-at} = ∫inf0 t2e-at dt

= ∫0inf t2e-(a+s)tdt

How do I integrate this? I tried using my TI-89 but it told me it is undefined. Any help would be greatly appreciated.

EDIT: The complete solution is required, which I think means I can't just take it out of the Laplace tables
 
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  • #2
Northbysouth said:

Homework Statement


Obtain the Laplace transformation of the function defined by

f(t) = 0 t<0

= t2e-at t>=0


Homework Equations





The Attempt at a Solution



I'm a little unsure of what I'm doing here, so bear with me.

L {t2e-at} = ∫inf0 t2e-at dt

= ∫0inf t2e-(a+s)tdt

How do I integrate this? I tried using my TI-89 but it told me it is undefined. Any help would be greatly appreciated.

EDIT: The complete solution is required, which I think means I can't just take it out of the Laplace tables

First of all: never, never write what you did above, which was
[tex] \int_0^{\infty} t^2 e^{-at} \, dt = \int_0^{\infty} t^2 e^{-(a+s)t} \, dt. [/tex]
This is only true if s= 0.

Anyway, for your final (correct) integral, change variables from t to x = (a+s)t (assuming s > -a).

I strongly recommend that you put away the TI-89 until after you have learned how to do these problems, or at least restrict its use to numerical computation.
 
  • #3
Sorry, I missed out a part.

There should have been an e-st

But I don't understand your point about changing the variable from t to x. Could you explain further please?
 
  • #4
Northbysouth said:
Sorry, I missed out a part.

There should have been an e-st

But I don't understand your point about changing the variable from t to x. Could you explain further please?

I assume you have taken integration already. If so, you already studied change-of-variable methods (or so I hope).

If you have not done integration yet you would need a longer explanation than I am prepared to give here: we would need to go over material that takes several weeks to present in coursework! However, such information is widely available in books and on-line.
 
  • #5
Ray Vickson said:
I assume you have taken integration already. If so, you already studied change-of-variable methods (or so I hope).

If you have not done integration yet you would need a longer explanation than I am prepared to give here: we would need to go over material that takes several weeks to present in coursework! However, such information is widely available in books and on-line.

Yes, it's just integration. But the trick you really need to handle the t^2 factor is integration by parts. By all means, review integration.
 
  • #6
It's much simpler to evaluate
[tex]\tilde{g}(s)=\int_0^{\infty} \mathrm{d} t \exp[-(a+s) t][/tex]
and then taking the 2nd derivative wrt. [itex]s[/itex]. It's very easy to see that
[tex]\tilde{f}(x)=\tilde{g}''(s).[/tex]
 

1. What is Laplace Transformation?

Laplace transformation is a mathematical technique used to convert a function of time into a function of complex variable. It is widely used in mathematics, physics, engineering, and other scientific fields to solve differential equations and analyze systems in the time domain.

2. How is Laplace Transformation different from Fourier Transformation?

The Laplace transformation and Fourier transformation are similar techniques, but they have different domains. The Laplace transformation is used to convert time-domain functions into complex frequency-domain functions, while the Fourier transformation is used to convert time-domain functions into real frequency-domain functions. This means that the Laplace transformation is more powerful because it can handle a wider range of functions, including those that do not have a Fourier transform.

3. What are the applications of Laplace Transformation?

Laplace transformation has many applications in various fields, including control systems, signal processing, circuit analysis, and probability theory. It is used to solve differential equations, analyze the stability of systems, and model complex systems in engineering and physics.

4. What are the advantages of using Laplace Transformation?

One of the main advantages of using Laplace transformation is its ability to convert differential equations into algebraic equations, making them easier to solve. It also allows for the analysis of complex systems in the frequency domain, providing insights into the behavior of these systems. Additionally, Laplace transformation can handle functions with discontinuities and is often more efficient than other methods of solving differential equations.

5. Are there any limitations of Laplace Transformation?

While Laplace transformation is a powerful tool, it does have some limitations. One limitation is that it requires the function to be transformed to have a finite number of discontinuities. Additionally, it may not be suitable for some functions with complex singularities, such as those with poles on the imaginary axis. Finally, the inverse Laplace transform may not exist for some functions, making it impossible to convert back to the time domain.

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