# Laplace Transform with Integral Convolution

1. Oct 31, 2013

### mechGTO

1. The problem statement, all variables and given/known data
Determine the Laplace Transform of
∫(from 0 to t) (t-τ)cos(2(t-τ))e-4τ

using Laplace Transform tables.

2. Relevant equations
I know the basic convolution theorem is
(f*g)(t) = ∫f(τ)g(t-τ)dτ

3. The attempt at a solution
I'm not sure if this is double convolution or if I'm missing an easy substitution.

2. Oct 31, 2013

### LCKurtz

1. What are $f(t)$ and $g(t)$ in this problem?
2. What is the formula for $\mathcal L(f*g)$?

3. Oct 31, 2013

### mechGTO

Response:
1. I'm not sure because I've never seen (t-tau) twice in one of these.
2. Laplace(f*g) = F(s)G(s)

4. Oct 31, 2013

### LCKurtz

As an example, what would $g(t-\tau)$ be for $g(t) = t^2\sin(t)$?

5. Oct 31, 2013

### mechGTO

In that problem, it would be setup so that f(t) = t^2 and g(t) = sin(t) making the convolution integral

∫ from 0 to t: t2*sin(t-τ) dτ

and then integrated using trig identities

Last edited: Oct 31, 2013
6. Oct 31, 2013

### LCKurtz

No. You didn't answer my question. I didn't ask you anything about convolution. Given $g(t) = t^2\sin(t)$ what would $g(\pi/2)$ be? $g(s)$? $g(t-\tau)$?

7. Oct 31, 2013

### mechGTO

$\int_0^t (t-\tau)cos(2(t-\tau))e^{-4\tau} d\tau$

it looks as if f(t) = t and g(t) = cos(2*f(t))

Would it make sense to do a simple substitution and end up doing the integral

$\int_0^t u\cdot cos(2u) du$

giving $\frac{1}{2} u\cdot sin(2u) + cos(2u)$

$\frac{1}{2} t\cdot sin(2t) + cos(2t)$

and then taking the laplace of those?

8. Oct 31, 2013

### mechGTO

I would say g(t-τ) would be (t-τ)2sin(t-τ)

9. Oct 31, 2013

### LCKurtz

No. You are missing the point of the problem. Just answer my question in my previous post for now. That will maybe give you an idea.

10. Oct 31, 2013

### LCKurtz

OK, so you see how you can have more than one $t-\tau$ for a function. Now can you identify $f(t)$ and $g(t)$ in your original problem?

11. Oct 31, 2013

### mechGTO

Yes, I do. Does that mean my attempt was incorrect?

12. Oct 31, 2013

### LCKurtz

If you mean your attempt in post #7, yes that is incorrect. Where did the exponential go in that post? But never mind that. What do you get for $f(t)$ and $g(t)$? Do you see how to do the problem knowing those and using post #3?

13. Oct 31, 2013

### mechGTO

ahh, got it. thanks a lot.

(sleep deprivation makes you miss things)