Laplace transform with unit step function

In summary, the Laplace transform of t H(t) is 1/s^2, but the book's answer is incorrect. To manipulate e^-t H(t) - e^-t H(t-1), use the property e-t = e-(t-1)e-1.
  • #1
Juanriq
42
0

Homework Statement


I'm trying to take the laplace transfrom of [itex] t H(t) [/itex] where H(t) is the unit step function. Also, in a separate problem I get [itex] e^{-t} H(t) - e^{-t}H(t-1) [/itex] and I am wondering how to manipulate it properly

Homework Equations

[itex] L \{ f(t-a) H(t-a) \} = e^{as}F(s) [/itex]



The Attempt at a Solution

For the first part, I thought that [itex] L \{ t H(t) \}[/itex] should just give [itex] \frac{1}{s^2} [/itex] back out, but the answer key in the book I'm using says that it is just [itex] \frac{1}{s} [/itex].



For the second part of my question, I know we have to manipulate the exponential, but how would I manipulate them? For instance, I want [itex] e^{t}H(t) [/itex], but can't I only multiply by a constant? Obviously [itex] e^{-t}e^{2t}H(t) [/itex] would do what I want, but now I'm introuducing something I can't factor outside the transform. Similarly for the second, [itex] e^{2t-1}e^{-t} [\latex] would do the trick... Any help is appreciated!
 
Physics news on Phys.org
  • #2
Your book is wrong about L[tH(t)]. Your answer is right.

For the second problem, use the fact that e-t=e-(t-1)e-1.
 

1. What is the Laplace transform with unit step function?

The Laplace transform with unit step function is a mathematical tool used to convert a function of time into a function of frequency. It is often used in engineering and physics to analyze systems and signals in the frequency domain.

2. How is the Laplace transform with unit step function defined?

The Laplace transform with unit step function is defined as the integral of a function multiplied by the unit step function, which is equal to 1 for all values greater than or equal to 0 and 0 for all values less than 0. The integral is taken from 0 to infinity.

3. What is the purpose of using the Laplace transform with unit step function?

The Laplace transform with unit step function allows us to analyze the behavior of a system in the frequency domain, which can provide insight into its stability and response to different inputs. It also simplifies the solving of differential equations, as it converts them into algebraic equations.

4. How is the Laplace transform with unit step function calculated?

The Laplace transform with unit step function can be calculated using the standard Laplace transform formula, with the addition of the unit step function term. This involves taking the integral of the function multiplied by the unit step function, and then applying any necessary properties or rules of integration.

5. What are some common applications of the Laplace transform with unit step function?

The Laplace transform with unit step function is commonly used in control systems, signal processing, and circuit analysis. It is also used in the study of differential equations and in the analysis of physical systems, such as electrical circuits and mechanical systems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
96
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
836
  • Calculus and Beyond Homework Help
Replies
1
Views
646
  • Calculus and Beyond Homework Help
Replies
3
Views
665
  • Calculus and Beyond Homework Help
Replies
3
Views
745
  • Calculus and Beyond Homework Help
Replies
1
Views
715
  • Calculus and Beyond Homework Help
Replies
1
Views
589
Back
Top