5.2-5.4, where you refine your powers of propositions and partial fractions

Section 5.2 is entitled “Basic Properties of the Laplace Transform.”

This section is just a ton of propositions. Seriously, I could leave the summary at just that and I think everyone would agree that I have successfully summarized this section.

Oh, well. Onto ALL of the definitions! (Note: Since this is just a summary blog, I will not be supplying the proofs. I shall be milking the “it’s just a summary” excuse quite a bit for this section.)

“Suppose y is a piecewise differentiable function of exponential order. Suppose also that y’ is of exponential order. Then for large values of s,

where Y(s) is the Laplace transform of y” (197).

The point of this proposition is that now it’s relatively easy to find the derivative of a function. Now, you could throw out really easy functions with really easy derivatives (f(t) = t, f(t) = sin(t), f(t) = e^t, etc.) but for other functions, it might be easier to use this method.

“Suppose that y and y’ are piecewise differentiable and continuous and that y” is piecewise continuous. Suppose that all three are of exponential order. Then

where Y(s) is the Laplace transform of y. More generally, if y and all of its derivatives up to order k-1 are piecewise differentiable and continuous, and y^(k) is piecewise continuous, and all of them have exponential order, then

“Suppose f and g are piecewise continuous functions of exponential order, and α and β are constants. Then

The point is that the Laplace transform of a linear combination of functions can be computed by taking the Laplace transform of each term separately and then adding up the result” (199).

“Suppose f is a piecewise continuous function of exponential order. Let F(s) be the Laplace transform of f, and let c be any constant. Then

“Suppose f is a piecewise continuous function of exponential order, and let F(s) be its Laplace transform. Then

More generally, if n is any positive integer, then

So that’s it for section 5.2. I told you it was a lot of propositions.

Section 5.3 is entitled “The Inverse Laplace Transform.”

Before we talk about the inverse, we need to introduce a theorem to clear up any concern.

“Suppose that f and g are continuous functions and that ℒ(f)(s) = ℒ(g)(s) for s > a. Then f(t) = g(t) for all t > 0” (203).

In other words, we have yet another uniqueness theorem to tuck underneath our belts. Also, I couldn’t prove this to you even if I had the proof in front of me, because apparently the proof goes beyond the scope of the book. That both terrifies me and excites me (I wouldn’t be a good physics major if I wasn’t curious about everything, now would I?).

Anyway, onto the definition:

“If f is a continuous function of exponential order and ℒ(f)(s) = F(s), then we call f the inverse Laplace transform of F, and write

Here’s a Paint picture to help clear up any confusion (and also for any visual learners out there):

A proposition for you: “The inverse Laplace transform is linear. Suppose that ℒ^-1(F) = f and ℒ^-1(G) = g. Then for any constants a and b,

Also, here’s a table of some common Laplace transforms (which you can probably find in a million other places here on the internet, but here it is nonetheless):

In order to complete this section, I have to bring back something I am not particularly fond of: partial fractions.

If you want a refresher on the subject (I know I did), here are a couple of websites for you (personally, I’m not very fond of the Wikipedia article on the subject. I’d say to go for the one with pretty colors): http://en.wikipedia.org/wiki/Partial_fraction_decomposition, http://www.purplemath.com/modules/partfrac.htm, http://www.mathsisfun.com/algebra/partial-fractions.html

The reason why this is brought up is because with the power of partial fractions, we can compute the inverse Laplace transform of most rational functions. I’ll include one example just to show you how it’s done!

That’s it for section 5.3! As long as you remember partial fractions, you really should be golden.

Section 5.4 is entitled “Using the Laplace Transform to Solve Differential Equations.”

In other words, the previous sections were more preface than anything else. They were merely the building the blocks for you to get to this section. Hooray! With your newfound power of partial fractions, Laplace transforms, inverse Laplace transforms, and propositions, you’ll be able to solve differential equations (with initial conditions, too)!

You can also solve higher order differential equations as well (say, order 4), and those equations are handled pretty much the same way. Anyway, here’s an overview of the method:

We’ll be looking at the following initial value problem:

We apply Laplace transform to this, where Y(s) = ℒ(y)(s):

If y is a solution to the initial value problem to our original differential equation, then ℒ(ay” + by’ + cy) = ℒ(f) = F. We can substitute this into the equation above and solve for Y. This means that

Something interesting to note about that final equation is that the denominator is the characteristic polynomial of our original differential equation.

Finally, two definitions:

Suppose y_s is the solution of

Also, suppose y_i is the solution of

Notice that although y_s has initial conditions equal to zero, it has the same forcing term (i.e. f(t)) as our original differential equation. This is referred to as the state-free solution. On the other hand, y_i is the solution to a homogeneous equation, but it has the same initial conditions. This is referred to as the input-free solution. From our previous derivations, we have

It’s pretty plain to see that Y can be written as Y = Y_s + Y_i, and from this we can see that y = y_s + y_i. Therefore, any initial value problems can be written as the sum of its state-free and input-free solutions.

And that’s it for 5.4 and the evening! 5.5 is going to be a bit more tedious than I previously thought, but it’s not that long of a section. I have a test in this class coming up fairly soon, and thus I would like to go about the test week the same way I did last time. I’d like to not have to summarize new material whilst trying to remember the old material.

Finally, a for-your-information type of thing, there are 3 more sections to cover in chapter 5, and then I’m going to ask about chapter 10 (in which we might look at the first two sections). If my teacher says I should summarize those as well, that means we have 5 more sections together before I’m off to bigger and better things (i.e. going onto Tumblr and spending these hours there).

Perhaps I will write an epilogue to this epic journey, just as a closer. We’ll cross that bridge when we come to it, though.

I’ll see you when I see you.