Finding general solutions using separation of variables
Addressing treating differentials algebraically
- [Instructor] So when you first learn calculus, you learn that the derivative of some function f, could be written as f prime of x is equal to the limit as, then there's multiple ways of doing this, the change in x approaches zero of f of x plus our change in x, minus f of x, over our change in x. And you learn multiple notations for this. For example, if you that y is equal f of x, you might write this as y prime. You might write this as d-y, d-x, which you'll often hear me say is the derivative of y, with respect to x, and that you can do the derivative of f with respect to x because y is equal to our function. But then later on when you, especially when you start getting into differential equations, you see people start to treat this notation as an actual algebraic expression. For example, you will learn or you might have already seen. If you're trying to solve the differential equation, the derivative of y with respect to x, is equal to y. So the rate of change of y with respect to x is equal to the value of y itself. This is one of, the most basic differential equations you might see. You'll see this technique, where people just say, "Well, let's just multiply both sides by d-x." Just treating d-x like as if it's some algebraic expression. So you multiply both sides by d-x and then you have, so that would cancel out algebraically, and so you see people treat it like that. So you have d-y is equal to y times d-x, and then they'll say, "Okay let's divide both sides by y." Which is a reasonable thing to do. Y is an algebraic expression. So if you divide both sides by y, you get one over y, d-y is equal to d-x. And then folks will integrate both sides to find a general solution to this differential equation. But my point on this video isn't to think about how do you solve a differential equation here, but to think about this notion of using, what we call differentials. So a d-x, or a d-y, and treating them algebraically like this. Treating them as algebraic expressions, where I can just multiply both sides by just d-x or d-y, or divide both sides by d-x or d-y. And I don't normally say this, but the rigor you need to show that this is okay in this situation, is not an easy thing to say. And so to just feel reasonably okay about doing this, this is a little bit hand wavy, it's not super mathematically rigorous. But it has proven to be a useful tool for us, to find these solutions. And conceptually the way that I think about a d-y, or a d-x, is this is the super small change in y, in response to a super small change in x. And that's essentially what this definition of the limit is telling us. Especially as Delta x approaches zero, we're going to have a super small change in x, as Delta x approaches zero. And then we're going to have a resulting super small change in y. So that's one way that you can feel a little bit better, of... And this is actually one of the justifications for this type of notation. As you can view this, what's the resulting super small, or what's the super small change in y, for a given super small change in x? Which is giving us the sense of what's the limiting value of the slope? As we go from the slope of the secant line to a tangent line? And if you view it that way you might feel a little bit better about using the differentials, or creating them algebraically. Let me just multiply both sides by that super small change in x. So the big picture is, this is a technique that you'll often see, in introductory differential equations classes. Introductory multi-variable classes and introductory calculus classes. But it's not very mathematically rigorous, to just treat differentials like algebraic expressions. But even thought it's not very mathematically rigorous to do it willy nilly like that. It has proven to be very useful. Now as you get more sophisticated in your mathematics there are rigorous definitions of a differential. Where you can get a better sense of where it is mathematically rigorous to use it and where it isn't. But the whole point here is, if you felt a little weird feeling about multiplying both sides by d-x or dividing both sides by d-x or d-y. Your feeling was mathematically justified, because it's not a very rigorous thing to do. At least until you have more rigor behind it, but I will tell you that if you're an introductory student it is a reasonable thing to do as you explore and manipulate some of these basic differential equations.
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