Laplace Transformations help me please?

In summary, the conversation was about finding Laplace transformations for various functions, including sin(sqrt(x)), cos(sqrt(x))/sqrt(x), erf(t)^(1/2), cos(x)/x, and e^(-x)/x. The conversation also mentioned using properties of Laplace transformations and integration methods to solve the problems. There was also a typo in the interval for problems (d) and (e), which was corrected. The conversation ended with a clarification that problem (d) is solving for the cosine integral and problem (e) is solving for the exponential integral.
  • #1
yusukered07
18
0
Laplace Transformations... help me please?

1. Homework Statement .

Find the laplace transformations of the following:

a. [tex]
\sin\, {\sqrt\,{x}}
[/tex]

b. [tex]
\frac{\cos\,{\sqrt{x}}}{{\sqrt{x}}}
[/tex]

c. [tex]
\ erf\,{(t)}^\frac{1}{2}}
[/tex]

d. [tex]
\int_{t}^\infty\;\frac{\cos\,x}{x}\
[/tex]

e. [tex]
\int_{t}^\infty\;\frac{{e}^{-x}}{x}\
[/tex]


2. Homework Equations .


3. The Attempt at a Solution


I used some properties of Laplace transformations, integration methods but still, didn't work.
 
Last edited:
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  • #2


are you sure about interval in prob d. and e.? I think it should be [t, inf].
 
  • #3


yeah... sorry typographical error.. It should be from t to infinity. Thanks!
 
  • #4


+++ For (d):
You have f(t) ==> f'(t) = -cost / t ==> tf'(t) = -cost
Now, find Laplace transform in both side and use final theorem, differentiation properties to find F.

+++ (e) is similar to (d)

p/s: trying to solve (a) :(
 
  • #5


d is called cosine integral right?

and e is exponential integral?
 
  • #6
Last edited by a moderator:

Related to Laplace Transformations help me please?

1. What is a Laplace transformation?

A Laplace transformation is a mathematical technique used to convert a function of time into a function of complex frequency, making it easier to solve certain types of differential equations.

2. What is the purpose of using Laplace transformations?

Laplace transformations are used to simplify the process of solving differential equations, particularly in engineering, physics, and other fields of science. They help to transform a complex differential equation into a simpler algebraic equation that can be easily solved.

3. How do I perform a Laplace transformation?

To perform a Laplace transformation, you need to apply the Laplace operator to the function of time. This involves integration, so you must have a good understanding of integration techniques and the properties of Laplace transformations.

4. What types of functions can be transformed using Laplace transformations?

Laplace transformations can be applied to a wide range of functions, including piecewise continuous functions, periodic functions, and functions with discontinuities. However, they are most commonly used for solving linear differential equations with constant coefficients.

5. What are the advantages of using Laplace transformations?

There are several advantages of using Laplace transformations, including simplifying the process of solving differential equations, reducing the amount of calculation required, and providing a more intuitive way of understanding the behavior of systems over time.

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