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University of North Texas

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Problem 5 Medium Difficulty

Show that each function is a solution of the given initial value problem.
Differential equation
$$y^{\prime}+y=\frac{2}{1+4 e^{2 x}}$$
Initial equation
$$y(-\ln 2)=\frac{\pi}{2}$$
Solution candidate
$$y=e^{-x} \tan ^{-1}\left(2 e^{x}\right)$$

Answer

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Video Transcript

we want to show that this initial value problem is a solution for our differential. Equus. Well, let's first go to fear. What? Why prime should be, Because then at that point, we just start plugging everything in. So they have why prime is equal to. So they take the derivative that we're gonna need to use, uh, product rule to start. So it would be negative. He to the negative action. Let me screw this over so we don't run out of room. So why primes, Eagleton? Negative e to the negative x tangent inverse of to heed the X. And then plus, now we take the derivative of tangent in verse, and that should be so first. That's right. This so it's gonna be one over. That's for Marie. You 1/1 plus u squared. So that should be won over. So war he to the two X and then we need to apply chain rule. So we take the derivative of our inside function, which is going to be to e to the X right now. Go Z to the X's cancel out there. So that's nice. Now what we want to do is use our initial value here and just start plugging everything. Get so let's local, the Senate and solve for y prime and see what we get. And then we can plug that value into our equation down there to see if we get the same thing. So this is gonna be why Prime is equal to to over one plus for he to the two x minus. Why now we're told that win. So this is X is equal to negative Natural log of two. Our output. Why should be pie? Don't Let's go ahead and plug that it. So we're gonna have to over one plus four. No, he to the negative natural log, too. Well, we can pull that negative to power inside, and I should be e to the natural log of one board. So now the e and natural log accounts out with each other and actually kind of keep on going, And then why should be high half now? Like I was saying, before the those counts out that we have one plus 1/4 which is five force to over by force minus pi uh, which should give us 8/5 minus pi out. So This is what we want to show why Prime is equal to. So now let's go ahead and plug this in the white prime. So why Prime is equal to So we want to plug in negative natural auger, too into here for all the exes. So let's see. So that will give us negative too Tangent in verse of now e to the negative selections to be 1/2. So we should have Tanja Inverse. Um, one plus then we have two times, Let's see one plus, Actually, that's the same thing that we had before. Or if we look up there so I should just be the or or 1/4 right here and now let's think so. Tangent inverse of one should be pie fourth, but we multiply that by negative. Tues. That's negative. Pie half. And then over here we would get the eight equals, but waas plus eight over. So that is we get why prime at negative. Natural too is equal to what we got when we solved or why Prime originally. So it looks like it does check out

University of North Texas
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