differentialequations-7

0:00—2:53 This is also written in the form of, it’s the K that on the right hand side, actually I found that sort of considerable difficulty and in general it is, these are the temperature concentration model. It’s nature pehabit K on the right hand side and to separate the arbitrary of qe is part of it. Another model for which that’s true is mixing also I think I will show you that on Monday. On the other hand, there is some common first order model which is not a natural way to separate things out. Examples will be the RC circuits, radio active buca, stuff like that. It’s not a universal utility but I thought that form of writing is a sufficient utility to make a special case and emphasis very heavily in the notes. Let’s look the equation this form will be good enough. When you solve it, let me remind you how the solution looks because that explains the terminology. The solution looks like act you’ve done the integrating factor, multiply through, and integrating both sides. You know, ensure you both to do. The solution looks like y equals, there is a term e to the negative k alpha front times an integral which you can make either definite or indefinite according to preference, q of t times e to the kt inside dt. It will help you to remember the opposite signs if you think q as a constant, 1 for example, these two guys to cancel out and produce a constant solution. That’s a good way to remember that the signs have to be opposite.

2:54—4:13 But I don’t encourage you to remember the formula at all, I am only, it’s just a convenient thing for me to be able to use right now. and there is another term which comes by putting out the arbitrary constant exclusively, it’s ce to the –kt. So you can either write this way, where this is somewhat vague, or you can make it definite by putting a 0 here, a t there, changing the dummy variable inside according to the way the note tells you to do it. Now, when you do this, and if K is positive, that’s absolutely essential. Only when that is so, then this term, I told me a few weeks ago, this term goes to zero because K is positive as t goes infinite. So this goes to zero as t goes, and it doesn’t matter what C is, to infinite. This term stays some sort of function, and so this term is called steady state, a long term solution, it’s called both. Long term solution.

4:14—5:33 And this, which appears to get smaller and smaller with time goes on, is therefore called transient because it disappears, as time goes to infinite. So this part uses the initial condition, initial value, let’s call it y(0), assuming you start the initial value when t is 0. It is a common thing to do though it’s not necessary. Staring value appears in this term, this one is just some function. Now, general picture the way that looks is the steady-state solution will be some solution, I don’t know, like that let’s say, so that’s the steady-state solution. What’s the other guy looks like? Well, the steady-state solution has this starting point, other solutions can have these other starting points, so in the beginning, they won’t look like the steady-state solution. But we know that as time goes on, they mush approach it because this term represents the difference between the solution and steady-state solution.

5:34—7:14 So this term goes for zero, and therefore whatever these guys do to start out with, after a while, they must follow the steady-state solution more and more closely. They must ensure to be anecdotic to it. So the solutions to any equations of that form will look like this. Even it is up here, maybe start at 127, that’s ok. After a while, it is going to start to approach that green curve. Of course they won’t cross each other but they are sort like, u know, that’s the rock star, and these are the groupies trying to get close to it. Now, but something follows from that picture, which is the steady-state solution? What is so special about this green curve? All these other white solution curves have that thing properly, the thing properly that all the other white curves and the green curve, too, all try to get closely to them. In other words, there is nothing about the green curve despite they all want to get close to it. Therefore there isn’t right to write a formula like that, there isn’t one steady-state, there are many. Now this produces vagueness. So you talk about the steady-state, the solution you talk about I have no answer to that, the answer is whichever one it looks simplest.