Laplace Transform of t^1 x e^(3t): Solving the Dilemma

In summary, the conversation discusses different methods for finding the Laplace transform of t^1 x e^(3t). The table being used points to the method for e^(at)f(t), but other methods such as using the basic formula for Laplace transform and the exponential property can also be used. All three methods will give the same result. The conversation also briefly mentions finding the Laplace transform of t^2 x e^(3t) and confirms that the same methods can be used.
  • #1
Ry122
565
2
Im trying to find the laplace transform of t^1 x e^(3t)
but looking it at the table, it looks like there's two different possible solutions for it.
one is for t^(n) x f(t)
and the other is for e^(at) x f(t)
which one do i choose?
 
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  • #2
Do you get different answers for both? In this case, the table I'm using would point to the one for e^(at)f(t).
 
  • #3
i do get two different answers.
doesnt ur table have t^(n) x f(t)? wouldn't that also satisfy it?
 
  • #4
My table doesn't have that one. What are your answers for each?
 
  • #5
this is what the table has.
http://users.on.net/~rohanlal/2222.jpg [Broken]
 
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  • #6
Okay, in your case n= 1 so it is just -F(s). What is F(s), the Laplace transform of e3t?

Of course, you could use the basic formula for Laplace transform:
[tex]L(s)= \int_0^\infty t e^{3t}e^{-st}dt[/tex]
using integration by parts.
 
  • #7
so i choose the other one for t^1 x e^(3t), the one that defender mentioned, because n = 1?
so if i used the one in my previous post, that would that be incorrect?
 
  • #8
You could also use the other property, namely that

[tex]\displaystyle \mathcal{L}[e^{at}f(t)] = F(s-a),[/tex]​

where [tex]F(s) = \mathcal{L}[f(t)].[/tex] Using this, you only need to get the Laplace transform of [tex]t[/tex], and evaluate it at [tex]s-3[/tex]. You should get the same result with both properties.

Good luck.
 
  • #9
The point is that all three methods:
a)[tex]\displaystyle \mathcal{L}[e^{at}f(t)] = F(s-a),[/tex]
where F(s) is the Laplace transform of t.

b)[tex]\displaystyle \mathcal{L}[tf(t)]= -F'(s)[/tex]
where F(s) is the Laplace transform of [itex]e^{3t}[/itex].

c)[tex]\displaystyle \mathcal{L}[te^{at}]= \int_0^\infty te^{(-s+3)t}dt[/tex]

will give the same result.

It would be a good exercise to try each method and see.
 
  • #10
for the laplace transform of t^2 x e^(3t) (n is greater than 1)
would http://users.on.net/~rohanlal/2222.jpg [Broken] be the correct one to use?
 
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  • #11
All three ways are correct, but I personally think the exponential property is the quickest, if you already know the Laplace transforms of polynomials.
 
  • #12
can you reread my previous post, i put in the wrong url for the image.
 

1. What is the Laplace Transform of t^1 x e^(3t)?

The Laplace Transform of t^1 x e^(3t) is equal to 1/(s-3)^2, where s is the complex variable used in the Laplace Transform. This can be derived using the standard formula for the Laplace Transform of t^n x e^(at).

2. Why is the Laplace Transform of t^1 x e^(3t) important?

The Laplace Transform of t^1 x e^(3t) is important because it allows us to solve differential equations involving this function using the Laplace Transform method. This method is often more efficient and easier to use than traditional methods of solving differential equations.

3. How is the Laplace Transform of t^1 x e^(3t) used to solve differential equations?

The Laplace Transform of t^1 x e^(3t) can be used to convert a differential equation into an algebraic equation, which can then be solved for the desired function. After obtaining the solution in terms of the Laplace Transform, the inverse Laplace Transform can be applied to obtain the final solution in the original domain.

4. What is the dilemma that arises when solving the Laplace Transform of t^1 x e^(3t)?

The dilemma that arises when solving the Laplace Transform of t^1 x e^(3t) is that the standard formula for the Laplace Transform of t^n x e^(at) is only valid for values of n greater than or equal to 0. Since t^1 has a negative power, this formula cannot be directly applied. This requires the use of additional techniques, such as partial fractions, to obtain the correct solution.

5. Are there any practical applications of the Laplace Transform of t^1 x e^(3t)?

Yes, the Laplace Transform of t^1 x e^(3t) has several practical applications in fields such as electrical engineering, control systems, and signal processing. It is commonly used to solve differential equations in these fields and is an essential tool for analyzing and designing systems with time-varying inputs and outputs.

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