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Current time:0:00Total duration:4:27

Worked example: exponential solution to differential equation

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.F (LO)
,
FUN‑7.F.1 (EK)
,
FUN‑7.F.2 (EK)
,
FUN‑7.G (LO)
,
FUN‑7.G.1 (EK)

Video transcript

so we've got the differential equation the derivative of Y with respect to X is equal to 3 times y and we want to find the particular solution that gives us Y being equal to 2 when X is equal to 1 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 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 going to I'm going to recognize that this is a separable differential equation and then I'm going to solve it that way so when I say it's separable that means we can separate all the Y's dys on one side and all the x's DX's on the other side and so what I could do is if I divide both sides of this equation by Y and multiply both sides by DX I get 1 over Y dy is equal to 3 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 equations now here on the left if I wanted to write it in a fairly general form I could write well the antiderivative of 1 over Y is going to be the natural log of the absolute value of y I'm taking the antiderivative with respect to Y here now I could add a constant but I'm going to add 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 antiderivative here is going to be is going to be 3x 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 3x plus C is equal to the natural log of Y I could write the natural log of Y is equal to e to the 3x plus C now I could rewrite this as equal to e to the 3x times e to the C now e to the C is just going to be some other arbitrary constant which I could still denote by C they're going to be different values but where do 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 3x 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 that either Y is equal to C e to the 3x or Y is equal to negative C II to the 3x well we've kept it in general terms I haven't put any we don't know what 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 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 2 when y is equal to du I'm not going to solve for C to find the particular solution X is equal to 1 or X when when X is equal to 1 Y is equal to 2 so I could write it like that and we get 2 is equal to C times e to the 3rd power 3 times 1 and so to solve for C I can just divide both sides by e to the third and so I could or I can multiply both sides times e to the negative 3rd and I could get 2 e to the negative third power is equal to C and so let's now substitute it back in and our particular solution is going to be Y is equal to C C is 2 e to the negative third power times e to the 3x now I have 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 2 times e to the 3x and then I'll add the exponents to the 3 X minus 3 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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