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Worked example: exponential solution to differential equation

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Video transcript

- [Teacher] So, we've got the differential equation, the derivative of y with respect to x is equal to three times y. And we want to find the particular solution that gives us y being equal to two when x is equal to one. So, I encourage you to pause this video and see if you can figure this out on your own. All right, now, let's work through it together. So, some of you might have immediately said, "Hey, this is the form of a differential equation "where the solution is going to be an exponential," and you just got right to it. But, I'm not going to go straight to that, I'm just gonna recognize that this is a separable differential equation and that I'm gonna solve it that way. So, when I say that it's separable, that means that we can separate the y's, dy's, on one side, and all the x's, dx's on the other side. And so, what I can do is if I divide both sides of this equation by y and multiply both sides by dx, I get one over y, dy, is equal to three-dx. Now, on the left and right hand sides, I have these clean things that I can now integrate. That's what people talk about when they say separable differential equation. Now, here on the left, if I wanted to write it in a fairly general form, I could write, well, the anti-derivative of one over y is gonna be the natural log of the absolute value of y. I'm taking the anti-derivative with respect to y, here. Now, I could add a constant, but I'm gonna add in a constant on the right-hand side, so there's no reason to add two arbitrary constants on both sides. I could just add one on one side. So, that is going to be equal to the anti-derivative here is going to be three-x and I'll add the promised constant, plus c, right over there. And now, let's think about it a little bit. Well, we can rewrite this in exponential form. We could say, we could write, that e to the three-x plus c is equal to the natural log of y. I could write the natural of y is equal to e to three-x plus c. Now, I could rewrite this is equal to e to three-x times e to the c. Now, e to the c is just gonna be some other arbitrary constant, which I could still denote by c. They're are going to be different values, but we're just trying to just get a sense of what the structure of this thing looks like. So, we could say this is going to be some constant times e to the three-x. So, another way of thinking about it. Saying the absolute value of y is equal to this. This isn't a function yet. We're trying to find this function solution to this differential equation. So, this would tell us either y is equal to c, e to the three-x, or y is equal to negative c, e to the three-x. Well, we've kept it in general terms. I haven't put any... We don't know c is. So, what we could do, instead, is just pick this one, and then we can solve for c assuming this one right over here. And so, we will see if we can meet these constraints using this and it'll essentially take the other one into consideration, whether we're going positive or negative. So, let's do that. So, when y is equal to two, I'm now going to solve for c to find the particular solution, x is equal to one, or when x is equal to one, y is equal to two. So, I could write it like that, and we get two is equal to c times e to the third power, three times one. And so, to solve for c, I could just divide both sides by e to the third, and so I could, or I could multiply both sides times e to the negative third, and I could get two e to the negative third power is equal to c. And so let's now substitute it back in and our particular solutions is gonna be y is equal to c. C is two-e to the negative third power times e to the three-x. Now, I'm taking the product of two things with the same base. I can add the exponents. So, I could say y is equal to two times e to the three-x and I'll add the exponents to three-x minus three, and there you go. This is one way that you could write the particular solution that meets these constraints for this separable differential equation.
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