# How to solve this problem using laplace transform?

• haha1234
In summary, the Laplace transform can be used to solve differential equations by converting them into algebraic equations that are easier to solve. This involves taking the derivative of the function, multiplying it by the Laplace variable, and integrating it. The result is a new function in the Laplace domain, which can then be transformed back into the time domain to find the solution to the original problem. This method is particularly useful for solving differential equations with initial conditions and non-homogeneous boundary conditions, making it a powerful tool in engineering and physics.
haha1234

## Homework Statement

The differential equation given:
y''-y'-2y=4t2

## The Attempt at a Solution

I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace transform of 1/s^3 and 1/s^2...

And the attached photo is the attempted solution.
https://www.physicsforums.com/attachments/92000

#### Attachments

• 20151118_133402[1].jpg
23.8 KB · Views: 588
haha1234 said:

## Homework Statement

The differential equation given:
y''-y'-2y=4t2

## The Attempt at a Solution

I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace transform of 1/s^3 and 1/s^2...

And the attached photo is the attempted solution.
https://www.physicsforums.com/attachments/92000
I didn't verify your work, but here is a table of Laplace transforms - http://web.stanford.edu/~boyd/ee102/laplace-table.pdf

haha1234 said:
But there is no conversion of 1/s^3 provided!
Look just below the one for 1/s2. It's a more general formula.

haha1234
haha1234 said:

## Homework Statement

The differential equation given:
y''-y'-2y=4t2

## The Attempt at a Solution

I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace transform of 1/s^3 and 1/s^2...

And the attached photo is the attempted solution.
https://www.physicsforums.com/attachments/92000

If you know the inverse transform of 1/s, then you can get the inverse transform of 1/s^2 by integration, and of 1/s^3 by integration again. Remember: there are some standard general transform results that are helpful. Below, let ## f(t) \leftrightarrow g(s) = {\cal L}(f)(s)##. Then:
$$\begin{array}{l} \displaystyle \frac{df(t)}{dt} \leftrightarrow s g(s) - f(0+)\\ \int_0^t f(\tau) \, d \tau \leftrightarrow \displaystyle \frac{1}{s} g(s) \end{array}$$
These were given specifically in the table suggested by Mark44; did you miss them?

## 1. What is the Laplace transform and how does it help in solving problems?

The Laplace transform is a mathematical technique used to transform a function from the time domain to the frequency domain. It allows us to solve differential equations and other problems involving complex functions by converting them into simpler algebraic equations, which can then be solved easily.

## 2. What are the steps involved in solving a problem using Laplace transform?

The first step is to take the Laplace transform of the given function. This involves integrating the function from 0 to infinity. The next step is to apply any necessary algebraic manipulations to simplify the transformed equation. Finally, the inverse Laplace transform is taken to obtain the solution in the time domain.

## 3. How do I choose the appropriate Laplace transform for a given problem?

The appropriate Laplace transform depends on the type of problem being solved. For example, if the problem involves a differential equation, the Laplace transform of the derivative would be used. It is important to understand the properties of Laplace transform and choose the appropriate form based on the given problem.

## 4. What are the advantages of using Laplace transform over other methods of solving problems?

The Laplace transform allows us to solve complex problems involving differential equations and other complex functions easily and efficiently. It also provides a graphical representation of the solution in the frequency domain, which can be useful in understanding the behavior of the system. Additionally, Laplace transform can be used to solve problems that are difficult or impossible to solve using other methods.

## 5. Are there any limitations to using Laplace transform in problem-solving?

Yes, there are some limitations to using Laplace transform. It is not applicable to all types of functions, and it may not always provide an exact solution. In some cases, the inverse Laplace transform may not exist, making it impossible to obtain the solution in the time domain. Additionally, the Laplace transform may not be suitable for problems with discontinuities or non-continuous functions.

• Calculus and Beyond Homework Help
Replies
10
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
410
• Calculus and Beyond Homework Help
Replies
1
Views
725
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
191
• Calculus and Beyond Homework Help
Replies
3
Views
909