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Show that $y=e^{-a t} \cos t$ is a solution of the differential equation $y^{\prime}=-e^{-a t}(a \cos t+\sin t) .$ Is this differential equation pure-time, autonomous, or nonautonomous?

Put the solution in the LHS and RHS of the differential equation and checkif LHS equals RHS. For detailed work, see the solution.

Calculus 2 / BC

Chapter 7

Differential Equations

Section 1

Modeling with Differential Equations

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

01:05

Verify that $y=-t \cos t-t…

03:10

Show that $y=\frac{2}{3} e…

00:33

Determine whether the diff…

01:24

Show that the solution of …

02:42

Verify that the given func…

01:48

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02:44

A differential equation is…

00:24

02:52

A differential equation A …

00:27

So we are has to figure out whether this equation e y equals e to the negative. 80 Times Co sign T were asked if that is a solution to why crime equals negative e to the negative 80 times a co scientist e plus sign t. So just remember solution means you plug it in both sides of the same. And this is a differential equation because it has the white prime that's a derivative. And it's actually a pure time differential equation because the only variable on the right side is t standing for time, as opposed to other types that you will study. So what we're going to do is take the derivative derivative of the top equation. They're using the product rule because it's too functions multiplied together and using the chain rule on this part because there's a function inside a function with the negative 80. Okay, so I'm living on to the next slide. If I have, why equals e to the negative 80 times co sign t. I have two functions. I've got my first function here. This is my F second function here. This is my G, and I'm gonna do that product rule F prime G plus f g prime. So my derivative is negative A e to the negative 80 Koh Sai Inti So there's my f prime right here using the chain role to get the negative and there's my G and then plus if times g prime So I have a negative sign of tea. So here's my f and then here is mine g prime. Now I just need to clean things up a little bit. So if you look at both of our chunks, both chunks have a negative. Both chunks have an E to the negative 80. See, you got a neat of the negative 80 and I've got a negative symbol. So I'm gonna do the distributive property here. So I have distributed I have a negative and I have an E to the negative 80. What was left over from the first chunk was an A and a co scientist E. And what was left over from the second chunk was a scientist E and that's it. I want to point out that here you could have taken the integral of this The anti derivative of that

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