# Systems Modeling DE Review

1. Apr 13, 2013

### cmmcnamara

Hi all, I'm taking a systems modeling class and as such a portion of the class consists of a differential equations review, mostly centered around Laplace transforms and some functions I've never had to deal with before so I was hoping for a bit of help if you'd all be so obliged! I have quite a few HW questions I can't seem to get just right.

The first one is to find the inverse laplace transform of:

$$f(s)=\frac{s+c}{(s+a)(s+b)^2}$$

The way I went about this one was to separate one of the factors of $\frac{1}{(s+b)}$ out which itself has its own reverse transform in the table I'm using. Then I recognized the other factor as the polynomial quotient form which also has its own transform. Since I then had the product of two transforms, the total transform should be the convolution of the two inverses. My answer is very close to what Wolfram Alpha has to say is the proper inverse transform, but I am off by a factor:

$$F(t)=\frac{(a-c)(e^{-bt}-e^{-at})}{(a-b)^2}+\frac{c-b}{a-b}te^{-bt}$$

I missing the factor of $e^{-bt}$ on the last term for some reason. I though I lost it during some portion of the convolution integral but I have triple checked my math and can't find anything.

Another problem I'm having is another missing factor in a problem finding the Laplace transform of a function:

$$F(t)=\mathcal{H}(t-5)(t-5)^4$$

I recognized that the binomial can be expanded and then the Heaviside function distributed to each term. This sets up a series of transforms of the form $(-1)^nt^nF(t)$ which has the transform of $f^{(n)}(s)$. In this case f(s) is the transform of the Heaviside function. I get the following answer after differentiating and simplifying:

$$F(t)=\frac{-24e^{-5s}}{s^5}(26s^4-1)$$

However Wolfram Alpha shows the proper answer is missing the factor of $26s^4-1$. Can anyone verify this as being correct/incorrect?

Those are the only real issues I'm having dealing with Laplace transforms. The other issues I am having revolve around solving ODE's that use the Heaviside and Dirac Delta functions as forcing functions. I have never had to deal with these before and am having some interesting troubles solving equations involving these functions. Is there any elementary reading sources on these functions besides Wikipedia? I'm finding their explanation of these functions are a bit over my head in terms of basic understanding. I'm attempting to solve them at the moment with Laplace transform as asked in my HW assignments but unless the initial conditions are 0 such that those terms cancel out, I'm finding the reverse transforms to be exceedingly difficult due to the additive terms (from the initial conditions) causing convolutions using the mentioned functions (which is at least the only way as far as I can tell) and those integrals have me scratching my head. Any help would be appreciated, thank you!

2. Apr 14, 2013

### HallsofIvy

Staff Emeritus
What did you get when you separated the integrand into partial fractions?