# Partial fractions (for laplace)

• bakin
In summary, the conversation is about solving a problem using partial fractions. The person is unsure about the next step and suggests using a new tool for solving such problems.
bakin

## Homework Statement

This problem is killing me.

I need to bust this thing up using partial fractions.

## The Attempt at a Solution

I'm leaning towards it being separated like this. Is this correct?

If it is, I'm not exactly sure what I'm supposed to do next.

Last edited:
145s/ [(s+2j)(s-2j)(s-2+3j)(s-2-3j)]
Make sure above is correct .. now

[145s / (s-2j)(s-2+3j)(s-2-3j) ]*1/ [(s+2j)]
+
[145s / (s+2j)(s-2+3j)(s-2-3j) ]*1/ [(s-2j)]
+
[145s / (s-2j)(s+2j)(s-2-3j) ]*1/ [(s-2+3j)]
+
[145s / (s-2j)(s+2j)(s-2+3j) ]*1/ [(s-2-3j)]

I believe what I did above should be self explanatory. Now next step is to substitute s=-2j to [145s / (s-2j)(s-2+3j)(s-2-3j) ]
s=2j to [145s / (s+2j)(s-2+3j)(s-2-3j) ]
s = 2-3j in the third, and s=2+3j .. so on, You leave one of the simplest possible factor in the denominator taking all others to the numerator and then substituting the root.

See if that works

I'll try that, thanks

bakin said:
I'll try that, thanks

Answering your question in the OP, you had that correct. But, I just wanted to share a new tool I discovered in Calc III class which I think is very helpful for partial fractions.

## 1. What are partial fractions in Laplace transforms?

Partial fractions in Laplace transforms are a method used to simplify complex rational functions into simpler components. This is done by breaking the rational function into a sum of smaller fractions with simpler denominators.

## 2. Why do we use partial fractions in Laplace transforms?

Partial fractions are used in Laplace transforms to make it easier to find the inverse Laplace transform, which is necessary to solve differential equations. By breaking the function into simpler components, the inverse Laplace transform can be found using a table of known transforms.

## 3. How do you find the partial fraction decomposition of a rational function?

To find the partial fraction decomposition of a rational function, you first factor the denominator into linear and irreducible quadratic factors. Then, you use the method of undetermined coefficients to find the coefficients of the partial fractions. This involves setting up a system of equations and solving for the unknown coefficients.

## 4. Can all rational functions be decomposed into partial fractions?

Yes, all rational functions can be decomposed into partial fractions as long as the degree of the numerator is less than the degree of the denominator. If the degree of the numerator is equal to or greater than the degree of the denominator, then the rational function is considered improper and must first be divided to make it proper.

## 5. What is the purpose of using partial fractions in Laplace transforms for solving differential equations?

The purpose of using partial fractions in Laplace transforms for solving differential equations is to make the process simpler and more efficient. By breaking the complex function into simpler components, the inverse Laplace transform can be found using a table of known transforms. This makes it easier to solve differential equations and obtain a solution in the time domain.

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