# Solve integral with laplace transform

• MHB
• goohu
In summary, the conversation discusses solving an integral with Laplace transform. The resulting expression is unclear and there is also uncertainty about the Laplace transform of y(t). The equation is then changed to take the Laplace transform and becomes a first-order linear differential equation. The solution is found to be Y(s)=2/s^2 + C/s^3 and after taking the inverse transform, y(t)=2t + Ct^2/2. The conversation ends with a question about applying the initial condition and checking the solution.
goohu
So the task is to solve the following integral with laplace transform.
View attachment 9423

Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)).

What I am unsure about is what happens with the integral part and how do we inpret the resulting expression?

What will it result in and how will be laplace transform the integral parts? I am also wondering what the laplace transform of y(t) will be.

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So you have the equation
$$3\int_0^t y(\tau)\,d\tau -t\,y(t)=t^2,\quad y(1)=3.$$
Here I've changed the variable of integration so it's less confusing. Now we take the Laplace Transform of the equation thus:
$$\frac{3Y(s)}{s}+Y'(s)=\frac{2}{s^3}.$$
This is now a differential equation in $s.$ It's first-order linear, so it should be pretty straight-forward to solve. Answer:
$$Y(s)=\frac{2}{s^2}+\frac{C}{s^3}.$$
Finally, the inverse transform yields
$$y(t)=2t+\frac{Ct^2}{2}.$$
Can you finish applying the initial condition, and checking that the solution works in the original integral equation?

## 1. What is the Laplace transform?

The Laplace transform is a mathematical operation that converts a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems.

## 2. How is the Laplace transform used to solve integrals?

The Laplace transform can be used to solve integrals by converting the integral into an algebraic equation in the complex frequency domain. This allows for easier manipulation and solution of the integral.

## 3. What are the steps for solving an integral with the Laplace transform?

The steps for solving an integral with the Laplace transform are as follows:

1. Take the Laplace transform of the integral.
2. Use algebraic manipulation to simplify the resulting equation.
3. Apply any known Laplace transform properties or theorems to further simplify the equation.
4. Take the inverse Laplace transform to convert the equation back into the time domain.

## 4. What are the advantages of using the Laplace transform to solve integrals?

The Laplace transform has several advantages for solving integrals, including:

• It can be used to solve a wide range of integrals, including those with complicated functions or boundary conditions.
• It can be used to solve both initial value and boundary value problems.
• It can simplify complex integrals into algebraic equations, making them easier to solve.

## 5. Are there any limitations to using the Laplace transform to solve integrals?

While the Laplace transform is a powerful tool for solving integrals, it does have some limitations. For example, it may not be applicable to integrals with discontinuous functions or those that do not have a finite limit as x approaches infinity. In addition, the inverse Laplace transform may not always have a closed-form solution, requiring the use of numerical methods for approximation.

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