Laplace & Inverse Laplace Transforms

In summary, the conversation discusses the Laplace transform of two functions, 1/(s^2+1)^2 + 1/(s^2+1) and ln(s+a). The first function can be calculated using a table of Laplace transforms while the second function does not have an elementary function as its Laplace transform. The conversation also mentions using the nth derivative in the Laplace transform.
  • #1
2RIP
62
0

Homework Statement


L[f(t)]= 1/(s^2+1)^2 + 1/(s^2+1)
L[f(t)]= ln(s+a) where 'a' is a constant

Homework Equations


The Attempt at a Solution


I know that the inverse laplace of 1/(s^2+1) is sin(t), but how do I deal with the squared form of it.

I have never encountered a logarithmic funcion for laplace, so can it be inverted back to f(t) with some of the common solution of conversion?

Thanks
 
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  • #2
2RIP said:

Homework Statement


L[f(t)]= 1/(s^2+1)^2 + 1/(s^2+1)
L[f(t)]= ln(s+a) where 'a' is a constant


Homework Equations





The Attempt at a Solution


I know that the inverse laplace of 1/(s^2+1) is sin(t), but how do I deal with the squared form of it.

I have never encountered a logarithmic funcion for laplace, so can it be inverted back to f(t) with some of the common solution of conversion?

Thanks
For the first problem, and using a table of Laplace transforms, I see that:
L(1/(2w^2)(sin (wt) - wt cos(wt)) = 1/(s^2 + w^2)^2
and L(sin(wt)) = w/(s^2 + w^2)

I'm stumped on the other problem
 
  • #3
No elementary function has ln(s+a) as its Laplace transform.
 
  • #4
f(t) = (-t)^n[f(t)]
F(s) = F(s)^nth derivative

I believe that's what I got to do for the second one. thanks
 

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 often used in engineering and physics to simplify differential equations and analyze systems in the frequency domain.

2. How is a Laplace Transform calculated?

A Laplace Transform is calculated using an integral formula that involves multiplying the function by an exponential term and integrating from 0 to infinity. The result is a new function of complex frequency.

3. What is the purpose of an Inverse Laplace Transform?

An Inverse Laplace Transform is used to convert a function of complex frequency back into a function of time. This is useful when analyzing systems in the frequency domain and then wanting to understand their behavior in the time domain.

4. How is an Inverse Laplace Transform calculated?

An Inverse Laplace Transform is calculated by using a table of Laplace Transform pairs or by using partial fraction decomposition and the inverse Laplace Transform formula. It is also possible to use software programs or calculators to calculate the inverse transform.

5. What are some common applications of Laplace and Inverse Laplace Transforms?

Laplace and Inverse Laplace Transforms are commonly used in engineering, physics, and mathematics for analyzing electrical circuits, control systems, and other dynamic systems. They are also used in signal processing, image processing, and fluid dynamics. Additionally, they have applications in probability and statistics, such as in the Laplace distribution.

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