# Laplace transforms

1. May 9, 2009

### Ry122

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?

2. May 9, 2009

### Defennder

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. May 9, 2009

### Ry122

i do get two different answers.
doesnt ur table have t^(n) x f(t)? wouldnt that also satisfy it?

4. May 10, 2009

### Defennder

My table doesn't have that one. What are your answers for each?

5. May 10, 2009

### Ry122

this is what the table has.
http://users.on.net/~rohanlal/2222.jpg [Broken]

Last edited by a moderator: May 4, 2017
6. May 10, 2009

### HallsofIvy

Staff Emeritus
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:
$$L(s)= \int_0^\infty t e^{3t}e^{-st}dt$$
using integration by parts.

7. May 10, 2009

### Ry122

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. May 10, 2009

### Metaleer

You could also use the other property, namely that

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

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

Good luck.

9. May 10, 2009

### HallsofIvy

Staff Emeritus
The point is that all three methods:
a)$$\displaystyle \mathcal{L}[e^{at}f(t)] = F(s-a),$$
where F(s) is the Laplace transform of t.

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

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

will give the same result.

It would be a good exercise to try each method and see.

10. May 10, 2009

### Ry122

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?

Last edited by a moderator: May 4, 2017
11. May 10, 2009

### Metaleer

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. May 10, 2009

### Ry122

can you reread my previous post, i put in the wrong url for the image.