Laplace Transform Practice: Finding Inverse and Basic Forms

In summary, the conversation discusses finding the Laplace transform of t sin(t) and t cos(t), as well as the inverse transform of \frac{1}{(1+s^2)^2}. The suggested method is to use convolution, and the hint is to consider the Laplace transform of \sin(t). A helpful table is also provided for quick reference.
  • #1
leopard
125
0

Homework Statement



Find the laplace transform of t sin(t) and t cos(t), and the inverse transform of [tex]\frac{1}{(1+s^2)^2}[/tex]

2. The attempt at a solution

I found the two laplace forms:

[tex]\frac{2s}{(s^2+1)^2}[/tex]

and

[tex]\frac{s^2-1}{(s^2+1)^2}[/tex]

I guess I'm supposed to use the two laplace transforms to find the inverse of this one, but I don't know how to do that.
 
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  • #2
You can try convolution... doesn't look hard. I partially solved it.
 
  • #3
Hint: what is the Laplace transform of [itex]\sin(t)[/itex]?
 
  • #4
leopard said:

Homework Statement



Find the laplace transform of t sin(t) and t cos(t), and the inverse transform of [tex]\frac{1}{(1+s^2)^2}[/tex]

2. The attempt at a solution

I found the two laplace forms:

[tex]\frac{2s}{(s^2+1)^2}[/tex]

and

[tex]\frac{s^2-1}{(s^2+1)^2}[/tex]

I guess I'm supposed to use the two laplace transforms to find the inverse of this one, but I don't know how to do that.

This is the best table I found if you need a quick answer in your research or to check your answer: http://www.vibrationdata.com/Laplace.htm.

For example check out 2.18, 2.20, and 2.21
 
  • #5
gabbagabbahey said:
Hint: what is the Laplace transform of [itex]\sin(t)[/itex]?

[tex]\frac{1}{s^2 + 1}[/tex]
 
  • #6
leopard said:
[tex]\frac{1}{s^2 + 1}[/tex]

Right, and so what is [tex]\mathcal{L}[\sin(t)-t\cos(t)][/tex]?:wink:
 
  • #7
Lol, it's so easy when you know the answer. One must be evil to give such a problem for the exam.
 

Related to Laplace Transform Practice: Finding Inverse and Basic Forms

1. What is a Laplace transform?

A Laplace transform is a mathematical operation that converts a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations.

2. Why is it important to practice finding inverse and basic forms of Laplace transforms?

Practicing finding inverse and basic forms of Laplace transforms helps improve mathematical skills and problem-solving abilities. It also allows for a better understanding of the fundamental concepts of Laplace transforms, which are frequently used in scientific research and real-world applications.

3. How do you find the inverse form of a Laplace transform?

The inverse form of a Laplace transform can be found by using the inverse Laplace transform formula or by using a table of Laplace transform pairs. The steps for finding the inverse form include simplifying the given function, using partial fraction decomposition if necessary, and then applying the inverse Laplace transform formula or table.

4. What are the basic forms of Laplace transforms?

The basic forms of Laplace transforms include the linearity property, the derivative property, the integral property, and the final value theorem. The linearity property states that the Laplace transform of a sum of functions is equal to the sum of their individual Laplace transforms. The derivative property states that the Laplace transform of a derivative of a function is equal to the Laplace transform of the original function multiplied by the variable s. The integral property states that the Laplace transform of an integral of a function is equal to the Laplace transform of the original function divided by the variable s. The final value theorem states that the limit of a function as time approaches infinity is equal to the limit of the Laplace transform of the function as the variable s approaches zero.

5. How can Laplace transforms be applied in real-world situations?

Laplace transforms have many real-world applications, such as in electrical engineering, control systems, and signal processing. They can be used to solve differential equations that model physical systems, analyze the stability of systems, and design controllers for systems. They are also used in financial mathematics, physics, and other fields.

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