# Laplace & Inverse Laplace transforms

• Cocoleia
In summary, the conversation is about a student's attempt at solving an equation using Laplace transforms. The student used partial fractions and their solution seemed simple, but the solution posted by their professor appears to have errors. The student is wondering if their solution is equivalent or if there are any faults in their logic and work. It is pointed out that there are indeed faults in the student's work, such as incorrect factorization and not recognizing that poles in the right-hand plane are problematic.
Cocoleia

## Homework Statement

I am given this equation:

and asked to solve using Laplace transforms

## The Attempt at a Solution

This is what I did:

This seemed logical to me, I used partial fractions and it stayed pretty simple.

This is what the solutions my prof posted do:

Is my answer equivalent to this, or are their faults in my logic and work ?

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One problem I can see with your work is where you write:

##(s^2 + 9) \neq (s - 3)(s + 3)##

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Also,
s^2 + 6s + 8 does not factor to (s-2)(s-4). Poles in the rt-hand plane are bad business!

Cocoleia said:

## Homework Statement

I am given this equation:
View attachment 218670
and asked to solve using Laplace transforms

## The Attempt at a Solution

This is what I did:
View attachment 218671
View attachment 218672
This seemed logical to me, I used partial fractions and it stayed pretty simple.

This is what the solutions my prof posted do:
View attachment 218673

Is my answer equivalent to this, or are their faults in my logic and work ?
Their (sic) are faults in your work ...

## 1. What is a Laplace transform?

A Laplace transform is a mathematical tool used to convert a function from the time domain to the frequency domain. It is represented by the symbol "L{ }" and is commonly used in engineering and physics to solve differential equations.

## 2. How do you perform an inverse Laplace transform?

An inverse Laplace transform is the reverse process of a Laplace transform and is used to convert a function from the frequency domain back to the time domain. It is represented by the symbol "L-1{ }" and can be performed using a table of Laplace transforms or through algebraic manipulation.

## 3. What is the purpose of using Laplace transforms?

Laplace transforms are used to simplify and solve differential equations, especially in the frequency domain where certain types of equations become easier to solve. They also allow for the analysis of systems in terms of their frequency response, making it useful in signal processing and control systems.

## 4. Can Laplace transforms be used for all types of functions?

No, Laplace transforms can only be performed on functions that are defined for all real numbers and have an exponential order. This means that the function must have a finite number of discontinuities and cannot grow faster than an exponential function.

## 5. Are there any limitations to using Laplace transforms?

One limitation of Laplace transforms is that they cannot be used for functions that have a singularity at the origin. They also cannot be used for functions that grow faster than an exponential function, as mentioned before. Additionally, numerical methods may be needed to perform inverse Laplace transforms for complex functions that do not have a known Laplace transform in the table.

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