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