# Solving ODEs with Laplace. Stuck at Partial Fraction Expansi

• CoolDude420
In summary: Well, is ##Q(sB+1) + Rs## equal to the constant ##A## for ALL values of ##s##? You need to figure out what values of ##Q## and ##R## make that happen. Did you really not see all this done in calculus 101?
CoolDude420

## Homework Statement

Hi,
So I had a pretty long question solving a Linear ODE but now I've gotten stuck at this stage where I can't seem to get it into the right form to carry out partial fraction expansion

## The Attempt at a Solution

[/B]
I'm quite sure that I what I have at the very last line isn't correct. I'm really new to solving ODEs with Laplace. Q and R are the constant things that you put over the fraction when solving with partial fraction expansion.

CoolDude420 said:

## Homework Statement

Hi,
So I had a pretty long question solving a Linear ODE but now I've gotten stuck at this stage where I can't seem to get it into the right form to carry out partial fraction expansion

## The Attempt at a Solution

View attachment 213056
[/B]
I'm quite sure that I what I have at the very last line isn't correct. I'm really new to solving ODEs with Laplace. Q and R are the constant things that you put over the fraction when solving with partial fraction expansion.
You have$$Y(s)=\frac{A}{s(sB+1)}$$You want to set that equal to its partial fractions like this:$$\frac{A}{s(sB+1)} = \frac Q s + \frac{R}{sB+1}$$
Add the two fractions on the right and compare numerators with the left to get ##Q## and ##R##.

LCKurtz said:
You have$$Y(s)=\frac{A}{s(sB+1)}$$You want to set that equal to its partial fractions like this:$$\frac{A}{s(sB+1)} = \frac Q s + \frac{R}{sB+1}$$
Add the two fractions on the right and compare numerators with the left to get ##Q## and ##R##.

Not sure how to compare these?

CoolDude420 said:
View attachment 213075

Not sure how to compare these?

Well, is ##Q(sB+1) + Rs## equal to the constant ##A## for ALL values of ##s##? You need to figure out what values of ##Q## and ##R## make that happen. Did you really not see all this done in calculus 101?

## 1. What is the Laplace transform and how does it relate to solving ODEs?

The Laplace transform is a mathematical tool used to solve differential equations. It converts a differential equation into an algebraic equation, making it easier to solve. This transformation is based on the concept of a complex integral and is particularly useful for solving initial value problems.

## 2. How does the partial fraction expansion method help in solving ODEs with Laplace?

The partial fraction expansion method is used to break down a complex rational function into simpler fractions. This is helpful in solving ODEs with Laplace because it allows us to rewrite the Laplace transform of a differential equation in terms of simpler functions, making it easier to solve and obtain the inverse Laplace transform.

## 3. What are the steps involved in solving ODEs with Laplace and partial fraction expansion?

The steps involved in solving ODEs with Laplace and partial fraction expansion are as follows:1. Take the Laplace transform of the differential equation.2. Use partial fraction expansion to rewrite the Laplace transform in terms of simpler functions.3. Transform back to the time domain using the inverse Laplace transform.4. Solve for the unknown function using algebraic methods.5. Check the solution by substituting it into the original differential equation.

## 4. What are some common challenges when solving ODEs with Laplace and partial fraction expansion?

Some common challenges when solving ODEs with Laplace and partial fraction expansion include:1. Finding the correct form of the partial fraction expansion.2. Dealing with complex roots and coefficients.3. Keeping track of the algebraic manipulations and simplifications.4. Ensuring that the solution satisfies any initial or boundary conditions.

## 5. Are there any limitations to using Laplace and partial fraction expansion to solve ODEs?

Yes, there are some limitations to using Laplace and partial fraction expansion to solve ODEs. These include:1. The method is only applicable to linear differential equations.2. It may not be effective for solving higher-order differential equations.3. The partial fraction expansion may not always be possible or straightforward to obtain.4. The method may become complicated for complex differential equations with multiple variables.

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