Laplace Transforms

i just took differential equations last term so i know how to do laplace transforms fairly well. however i dont know how to teach someone math over the internet. i guess if you have any specific questions just ask me.

I understand the beginning at least a little bit. I can solve IVP's and stuff with the laplace transform. Once it gets to step functions I get confused as hell though. I mostly just don't get what it means. Like when I see Uc(t)f(t-c), I realize c is replaced by some integer...but I don't get what that means.

Or say this: (might be hard to do in text..)

(I used _< to mean less then or equal)
f(t) = sin(t) from 0 _< t < pi/4
= sint(t) + cos(t-pi/4) from t >_ pi/4

And then the book just says, by that data we know g(t) is equal to:

g(t) = 0 from t < pi/4
= cos(t-pi/4) from t >_ pi/4

And so we know that:

g(t) = U(pi/4)(t)cos(t-pi/4)


Sorry that is probably really confusing to understand, I'll try and think of a different way to show it. However, if you can decipher it, I guess I don't understand how they can just jump to the next step. Like where do they get the g(t) data from? And then I don't get how the whole Uc(t) ties into it.

Ok, thats pretty confusing. I think what you need to do is use the formula L(h(t-a)f(t-a))= e^(a*s)L(f). You need to manipulate the shit you've been given (by using algebra, trig, and diff eq.'s) to get it into the form on the left side so you can simplyfy it. You don't really need to know why, it's just a basic formula that I dont know the proof to. I might be totally off from what you needed but i hope that helps a little bit.

Ok, I have another question for you. Say I'm taking the inverse of the laplace transform of:

[e^(-pi/2))(s)] * [(1/s)-s/(s^2+1)]

Now I know that the first part, is going to give me like:

Uc(t) where c = pi/2. That was separated from equations earlier.

When I take the inverse of the second part, I get basically:

1-cos(t)

However, in my book, the stuff that the Uc(T) is multiplied by has to be taken from (t-pi/2), or I assume (t-c) where I know my c value. Why do I have to put in the (t-c) part? That turns it in to this:

1-cos(t-c) = 1-sin(t) by trigonometric identities.

This gives me a final answer of Uc(T)*[1-sin(t)]

So in summary, if you are still with me haha..why do I have to change it from (t) to (t-c). Any ideas?

its just the identity.... if it tells you it has to be (t-c) then you need to do that. in your case it does turn out to be pretty simple to convert because its just by a factor of pi/4. just follow all of the given formulas, because you know those are right, and work your way down from there.

I'm not sure how much I'm helping but i have to go do some stuff now, if you want to post some more questions I'll respond later today, if not good luck man.

We are doing this shit too, i dont get it whatsoever

Ya man it sucks. I'm slowly figuring it out, but I have a test on it on tuesday so I think either way i'm probably fucked.

After taking diff eq I promptly forgot just about everything I learned, especially Laplace Transforms.

I'm slowly figuring them out....at this point in time though, I see no application towards me ever using them and all I can really think to say is "Fuck this shit." I hate calc haha.