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That might seem like a sort of fussy thing to talk about in the third or fourth lecture, but the reason is that so far, you know how to solve two kinds of differential equations, two kinds of first-order differential equations, one where you can separate variables, and the linear equation that we talked about last time.
Now, the sad fact is that in some sense, those are the only two general methods there are, that those are the only two kinds of equations that can always be solved. Well, what about all the others?
The answer is that to a great extent, all the other equations that can be solved, the solution can be done by changing the variables in the equation to reduce it to one of the cases that we can already do.
But, ultimately, as you will see, the way the equations are solved is by changing them into a linear equation, or an equation where the variables are separable. The first change of variables that I want to talk about is an almost trivial one.
So, in that sense, the most common change of variables is the one simple one called scaling. And, scaling simply means to change the coordinates, in effect, or axes, to change the coordinates on the axes to scale the axes, to either stretch them or contract them.
So, what does the change of variable actually look like? Well, it means you introduce new variables, where x1 is equal to x times something or times a constant. And y, the same. So, the new variable y1 is related to the old one by an equation of that form. So, a, b are constants.
So, those are the equations. Why does one do this? Well, for a lot of reasons. But, maybe we can list them. You, for example, could be changing units.
Changing the units that he used, you would have to make a change of coordinates of this form. In other words, so that the variables become pure numbers, with no units attached to them.
It just represents the number three, not three seconds, or three grahams, or anything like that. And, the third reason is to reduce or simplify the constants: Reduce their number is self explanatory. Let me give you a very simple example which will illustrate most of these things.
So, T is the internal temperature, the thing we are interested in. Now, you are free to solve that equation just as it stands, if you can. But, before you do that, one should scale.
How should I scale? This is now dimensionless because M, of course, has the units of temperature, degrees Celsius, degrees absolute, whatever it is, as does T. And therefore, by taking the ratio of the two, there are no units attached to it. So, this is dimensionless.
Now, how actually do I change the variable in the equation? Instead, do the inverse. The new equation will be what? For one, I think it looks better. Notice, so, T1 has been rendered, is now dimensionless. How you are measuring temperature makes no difference to this equation.
What units does it have? So, k1, whereas it had units involving both degrees and seconds before, now it has inverse time as its units. So, one less constant in the equation. It just looks better. This business, I think you know that k1, the process of forming k1 out of k M cubed is called lumping constants.
I think they use standard terminology in physics and engineering courses. Try to get all the constants together like this.MIT OpenCourseWare benjaminpohle.com Differential Equations, Spring Transcript – Lecture 12 We are going to start today in a serious way on the.
Students should strive for personal mastery over the following skills. These are the skills that are used in other courses at MIT. This list of skills is widely disseminated among the faculty teaching courses listing as a prerequisite.
At the moment, courses at MIT list as a prerequisite or a corequisite. Differential Equations are the language in which the laws of nature are expressed.
Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. MIT Course Number. As Taught In. Spring Level. Undergraduate.
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» MIT OpenCourseWare» Mathematics» Differential Equations, Spring Lecture Notes These lecture notes are pdf renderings of simple text files that were provided to the students containing nearly verbatim transcripts of Professor Miller's lectures.
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