1. ## Laplace Transforms

Hi,

For Question 2,

I know you can do them by partial intergration but I'm not sure how to show them using the method it asks.

For Question 4, I genuinely have no idea.

Any help would be appreciated.

Thanks

2. ## Re: Laplace Transforms

Originally Posted by fourierT
Hi,

For Question 2,

I know you can do them by partial intergration but I'm not sure how to show them using the method it asks.

For Question 4, I genuinely have no idea.

Any help would be appreciated.

Thanks

By "using the definition" they mean

$\large F(s) = \mathscr{L}\left \{f(t) \right \} = \displaystyle{\int_{-\infty}^{\infty}}f(t)e^{-st}~dt$

and for 2(a) you would use "integration by parts" to evaluate this integral.

for 2(b) just use their hint and do the integration.

I should note that it looks like they implicitly mean f(t)=0 for t<0, otherwise these transforms won't converge.

for 4) you should have read about using Laplace transforms to solve linear constant coefficient differential equations. This is a pretty straightforward example assuming H(t) stands for the Heaviside step function. Go re-read that section of your text. Or look at this.

3. ## Re: Laplace Transforms

Done the rest, still stuck on Question 4.

Any further help would be great, thanks.

4. ## Re: Laplace Transforms

Originally Posted by fourierT
Thanks,

I'm slightly behind on the laplace stuff hence why I had a some trouble with these.

I think I've done 2 a) and b) but still can't do 4).
If we have two transform pairs $f(t) \overset{\mathscr{L}}{\Longleftrightarrow} F(s)$

Then what does $\dfrac{d}{dt}f(t)$ correspond to in the s domain?

5. ## Re: Laplace Transforms

sF(s) - F(0) ?

6. ## Re: Laplace Transforms

No, that doesn't even make sense - derivative of F(s) with respect to t?

The Laplace transform of f'(t) has an easy-to-write relationship to the Laplace transform of f(t). It should be in your textbook, or you can look at romsek's link.

- Hollywood

7. ## Re: Laplace Transforms

Originally Posted by fourierT
sF(s) - F(0) ?
ok now apply this twice and take the Laplace transform of both sides of your differential equation and solve it in the s domain. Then transform it back to the t domain.