# Solving Laplace Transform: L{texp(9t)sin(2t)}

• Pete_01
In summary, to find the Laplace transform of texp(9t)sin(2t), use the definition of Laplace transform and integration by parts. Set u = t and dv = exp(9t)sin(2t)dt, then use another integration by parts to find v.
Pete_01

## Homework Statement

Find the Laplace transform of L{texp(9t)sin(2t)}

(-1)^n F^n (s)

## The Attempt at a Solution

I am not sure how to start it. The exp(9t) term throws me off. Usually I would just do (-1) F'(s) where F(s) is the laplace of sin(2t). But do I take the derivative of exp(9t) in addition to sin(2t)? Multiply them together?

You can start by using the definition of Laplace transform:
$$\int_0^\infty f(t)e^{-st}dt$$

Here, $f(t)= t e^{9t} sin(2t)$ so that integral is
$$\int_0^t te^{(9- s)t} sin(2t) dt$$

Integration by parts should do that. The obvious choice is u= t, $dv= e^{(9-s)t}sin(2t)dt$ and you will have to do another integration by parts to find v.

## 1. What is a Laplace transform?

A Laplace transform is a mathematical operation that transforms a function of time into a function of complex frequency. It is commonly used to solve differential equations and is an important tool in many areas of science and engineering.

## 2. How do you solve a Laplace transform?

To solve a Laplace transform, you first need to apply the Laplace transform formula to the given function. Then, you can use tables or algebraic techniques to simplify the transformed function and find the solution. It is important to follow the proper steps and techniques to ensure an accurate solution.

## 3. What is the purpose of solving a Laplace transform?

The main purpose of solving a Laplace transform is to simplify and solve differential equations. It is also useful in analyzing systems and signals in engineering and physics. Additionally, it can be used to solve initial value problems and boundary value problems.

## 4. Can you solve a Laplace transform for any function?

No, not all functions have a Laplace transform. The function must be of exponential order, meaning that it must decrease faster than any exponential function as the variable approaches infinity. Additionally, it must be piecewise continuous, meaning that it cannot have any abrupt changes in value.

## 5. What are some common applications of Laplace transforms?

Laplace transforms have many applications in science and engineering. They are commonly used in control theory, where they help analyze and design systems that respond to inputs. They are also used in signal processing, circuit analysis, and fluid dynamics, among others.

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