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Laplace transforms

  • Thread starter Ry122
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  • #1
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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?
 

Answers and Replies

  • #2
Defennder
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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
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i do get two different answers.
doesnt ur table have t^(n) x f(t)? wouldnt that also satisfy it?
 
  • #4
Defennder
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My table doesn't have that one. What are your answers for each?
 
  • #5
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this is what the table has.
http://users.on.net/~rohanlal/2222.jpg [Broken]
 
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  • #6
HallsofIvy
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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
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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
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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
HallsofIvy
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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
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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
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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
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can you reread my previous post, i put in the wrong url for the image.
 

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