First allow me to voice my hatred for differential equations! That being said, I need help with some initial value problems using delta impulses in step functions.

to start the problem reads

y''-7y'-8y=d(t-5)

I took the laplace of both sides and got
Y(s^2-7s-8)=e^(-5s)

simplified to

Y=[e^(-5s)]/[(s+1)(s-8)]

Now I need to take the inverse Laplace transform of this, using the notation Uc

I am stuck on the inverse laplace. Any pointers or help?

Your next step will be partial fractions.

[A/(s+1)] + [B/(s-8)] = 1

In trying to remember how to do the dirac delta, I found an awesome link that should help you out: http://www.math.psu.edu/srikrish/math251/notes/diracdelta.pdf

There's some good examples in there very similar to what you're working with. Good luck.

Fuck diff-eq.

That's all I have to say.

I made a concentrated effort to kill those brain cells after college.

Your next step is NOT partial fractions. Don't listen to that.

Here is information fresh out of my Dynamic Systems and controls class.

Let's see: Your transfer function is Y(s)=e^-5s/((s+1)(s-8))

You have an unstable system with poles of s=-1 and s=8.

The trick to solving problems with a time delay (e^-theta*s), is that you ignore the time delay until the end. Your theta is 5. So let's ignore the time delay term. What is the inverse Laplace of 1/[(s+1)(s-8)]? First you have to realize it is not in standard time constant form. In standard time constant form, the transfer function looks like: 1/[(s+1)*(-1/8*s+1)]. Your first time constant is +1. Your second time constant is -1/8. The inverse Laplace of an equation of this form is: 1/(b1-b2)*(exp(-b2t)-exp(-b1t)).

The result is 1/(1- -8)*(exp(--8*t)-exp(-1*t)) or 1/9*(exp(8t)-exp(-t)).

Now for the time delay. Since your theta is 5, anywhere you see a t in the final equation y(t), put in (t-5). Your resulting equation becomes y(t)= 1/9*(exp(8*(t-5))-exp(-(t-5))).

EDIT: Took out standard time constant form because it gets really confusing, I just did that and would have to explain much more.

Feel free to pm me with questions. I will also try to check back on this thread.

Time delay, transfer function, etc, all terms not used in a DE class. That is dynamic systems and controls territory. There's a good chance that, since OP is asking for ODE help, he has not taken such a class. Since you have, I would ask you to reach back and remember what a dirac delta function is. The problem format, OP asking for Uc notation, and the transfer function itself all indicate dirac delta.

By this point in your education, you should know there's more than one way to skin a cat, so hesitate before calling out someone else as wrong.

Do they have calculators now that differentiate?

I don't know. I tried to use one of the ti 89's one time, got pissed after 10 minutes, gave it back to my roommate. I'm glad I never bought one. Most people that use those sorts of calculators fail to understand what is happening, and miss the point of the class.

Differential equations may have more than one answer.

Even the form of the answer can vary based on the bais equations used in the solution.

The Fourier and Laplace equations have a whole lot of assumptions 'hiding' in them. that are routinely ignored.

It is one of those things that if the answer works under the assumptions, it is valid.

LOL thanks for all the help guys,  I ended up going to the tutoring center, and they got me squared away.

I passed the class last semester

I barely passed the class but I studied a ton and can say that I have a solid graps of the concepts, but the actual formulas i still reference in the book or online.

And yes, as others have said, Fuck Diff EQ!