# Another Laplace Transform Question

• jumbogala
In summary, the Laplace transform of f(t) is -9/s^2 + 7/s + e^{5s}(-2/s + 9/s^2 + 1/s). The step function u5(-7 + 9t + et) is handled by using the formula L[u_c(f(t))] = e^{-cs}L[f(t+c)].
jumbogala

## Homework Statement

Find the Laplace transform of f(t).

f(t) = -9t + 7 + u5(-7 + 9t + et)

## The Attempt at a Solution

The Laplace transform of -9t is just -9/t2

The Laplace transform of 7 is 7/s

But I am not sure how to find it for u5(-7 + 9t + et). In the tables it says ua(t)f(t-a) = e-asF(s). Is that the line I should use? Even if it is, I don't know how.

You've got the right start. You want to take it term by term, and you have the Laplace transform down for the first two terms. The way to handle step functions is as follows. First, get rid of the step function by saying:

$$L[u_c(f(t))] = e^{-cs}L[f(t+c)]$$

So in your case, the third term would become

$$e^{-5s} L[-2 + 9t + e^t]$$

Now it's just another term by term Laplace transform. So your final answer should be

$$-9/s^2 + 7/s + e^{5s}(-2/s + 9/s^2 + 1/s)$$

n.b.: Remember that the Laplace transform is always a function of s, while the inverse Laplace transform is a function of t. I see you got it slightly mixed up, so just something to keep in mind! :)

## 1. What is the Laplace Transform?

The Laplace Transform is a mathematical tool used to solve differential equations and analyze dynamic systems. It transforms a function from the time domain to the frequency domain, allowing for easier analysis and solution of complex systems.

## 2. How is the Laplace Transform calculated?

The Laplace Transform is calculated by integrating a function multiplied by the exponential function e^-st, where s is a complex variable. The result is a new function in the frequency domain.

## 3. What is the significance of the Laplace Transform in science?

The Laplace Transform is a powerful tool in various fields of science and engineering. It is used to analyze and model complex systems such as electrical circuits, mechanical systems, and biological processes. It also has applications in control theory, signal processing, and image processing.

## 4. Can the Laplace Transform be used to solve any type of differential equation?

No, the Laplace Transform can only be used to solve linear differential equations with constant coefficients. Non-linear and variable coefficient differential equations require other methods for solution.

## 5. How is the Laplace Transform related to the Fourier Transform?

The Laplace Transform is a generalization of the Fourier Transform. While the Fourier Transform is used for periodic functions, the Laplace Transform can be applied to non-periodic functions as well. In fact, the Fourier Transform is a special case of the Laplace Transform when s=0.

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