1. ## Laplase Transformations! Please save me from this problem!

Find the laplace transformation of the following:

2. Originally Posted by Bananna
Find the laplace transformation of the following:
I have some thoughts but I don't have time right now. If no-one else jumps in I might have time later. If you're self-instructing, where are these questions coming from?

3. A study guide, from my math tutor, I have a teacher but he is incomprehensible! I will appreciate any help I can get I have a test on Thursday night and my anxiety level is way too high, because all the problems I am having with this study guide.

4. Originally Posted by Bananna
Find the laplace transformation of the following:
Again I will only give hints:

You better do this using the definition of Laplace Transform.

To do it using properties you need three things

1) $L[e^{-3t}] = \delta(t - 3)$

2) $L[\int_{0}^{t} y(\tau) d\tau] = \frac{L[y(t)]}{s}$

3) Trickiest one $L[\frac{\sin 2t} {t}]$. This is the sinc function.Trying this separately is a good idea

5. Originally Posted by Isomorphism
Again I will only give hints:

You better do this using the definition of Laplace Transform.

To do it using properties you need three things

1) $L[e^{-3t}] = \delta(t - 3)$

2) $L[\int_{0}^{t} y(\tau) d\tau] = \frac{L[y(t)]}{s}$

3) Trickiest one $L[\frac{\sin 2t} {t}]$. This is the sinc function.Trying this separately is a good idea
Isomorphism has given some good hints. I'll give some different ones:

1. $LT \left[ e^{at} f(t) \right] = F(s - a)$ where F(s) = LT[f(t)]. This is one of the famous shift theorems.

In your problem, a = -3 and $f(t) = \int_0^t \frac{\sin (2\tau)}{\tau} \, d \tau$.

2. Same as Isomorphism's hint. In your problem $y(t) = \frac{\sin (2t)}{t}$.

3. $LT\left[ \frac{g(t)}{t} \right] = \int_{s}^{\infty} G(\tau) \, d \tau$ where G(s) = LT[g(t)]. In your problem $g(t) = \sin(2t)$.

4. $LT[\sin (2t)] = \frac{2}{s^2 + 2^2}$.

5. Therefore $LT\left[ \frac{\sin (2t)}{t}\right] = \int_{s}^{\infty} \frac{2}{\tau^2 + 2^2} \, d \tau = .....$

I will point out that all of these results should be known to you. I can't help wondering if the learning curve of your self-study is too steep. Have you gone through - thoroughly - the basics?

6. Originally Posted by Isomorphism
[snip]

3) Trickiest one $L[\frac{\sin 2t} {t}]$. This is the sinc function.Trying this separately is a good idea
Unfortunately, it's not the sinc function. It's actually 2 sinc(t) cos(t) ...... It's best handled using the result I give in hint 3.

7. Originally Posted by mr fantastic
Unfortunately, it's not the sinc function. It's actually 2 sinc(t) cos(t) ...... It's best handled using the result I give in hint 3.
I was not explicit since it was a hint:
$L[\frac{\sin 2t}{t}] = 2L[\text{sinc}(2t)]$
Either a change of variable when solving by basics OR the time compression property would have finished the job

8. Originally Posted by Isomorphism
I was not explicit since it was a hint:
$L[\frac{\sin 2t}{t}] = 2L[\text{sinc}(2t)]$
Either a change of variable when solving by basics OR the time compression property would have finished the job
Yes, I thought of 2 sinc(2x) later

But, I think getting it's transform from basics would be difficult for this member. And I don't think the transform of sinc (x) would be on the tables this member is using, which would make the time compression property unusable. Using the appropriate operational theorem is the most expeditious approach, I think.

What would be interesting is to see what tables this member will be using in the exam ......