WEBVTT
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All right. So we have a differential equation.
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Second order. Why? Double crime is equal.
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Thio co sign of T minus y and then plus
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y squared. And we have some initial conditions in
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the initial conditions. Are that why? So here
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, wise, of course. A function of tea
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. And if we evaluate why at syrup we get
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one. And if we evaluate why prime and zero
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, we get zero. And what we would like
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to do is convert this differential equation into a system
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. Uh, first order equations, OK, And
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the way you do that is really simple. So
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you just say x once and my first variable is
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just going to be Why in my second variable is
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going to be why prime? And we only have
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to do this up into until one less than the
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derivative that gives the order of the differential equation.
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So in this case, since it's the second order
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, we only need two variables. And now notice
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that we have one equation that's really easy to see
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, So we have that X one prime. That's
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just why prime is equal to X to. So
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this is one of our for shorter equations And then
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our next equation is gonna come from our original equation
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that why double prime is X to crime and then
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we have co sign it's okay to have a team
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, but why? We want to write his ex
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one. And then again we have x one square
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. So what we just did is we converted this
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first or the second order differential equation into a system
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of two first word or equations and now our conditions
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on ex one next to are the X one,
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which is why zero is equal to one and then
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x, too, Which is why prime of zero
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equal zero. So here is what's called the standard
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form of this system of differential equations.