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Hey there, in this video we'll recall how to solve a few different kinds of differential equations.
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Specifically, we'll look at the three equations listed here.
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The key to solving any differential equation is first figuring out what kind of differential equation it is, so you know what kind of method you use to solve it.
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This means that at all times we should have a working knowledge of the kinds of differential equations that can show up so that we know what we're looking for.
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Now starting out, you usually run into four different kinds of differential equations.
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There are separable equations which can be expressed as a function of y times dy equals a function of x times dx.
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There are a linear equation which can be expressed as dy equals f of x times y plus g of x all times dx.
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There are homogeneous equations which arrive in the form dy equals a function of y over x, dx.
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And finally there are exact equations which are of the form of function of x and y, p of x, y, x plus another function of xy, q of xy times d y equals 0 with the constraint that the partial derivative of p with respect to y is equal to the partial derivative of q with respect to x and just as a quick point of caution there's also a notion of a homogeneous linear differential equation which is different from a homogeneous equation so there are two different kinds of homogeneous homogeneous versus homogeneous linear so just be on the lookout for that now we have three different equations above and four different options.
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And it'd be a great idea right now to pause the video and try to figure out which of the equations above fits which of these kinds of differential equations.
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And remember, you might have to do a little bit of algebraic manipulation in order to get one thing into the correct form.
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Looking at that first equation, if you decided that it was an example of an exact equation, then you're right.
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Here the first term in parentheses is my p of xy value.
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The second term is my q of xy value.
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Value and it's very easy to check that the partial derivative of p with respect to y is 2xy plus 1 and likewise the partial derivative of q with respect to x is 2xy plus 1 so these two things are equal and this is an exact equation now the way that we solve an exact equation is by integrating specifically what i want to do is i want to find a function f of xy so the partial derivative of f respect to x is p and the partial derivative of f with respect to y is q take either one of these equations and integrate both sides.
02:32
However, when i'm doing integrals in this particular context, i need to remember that i'm doing partial derivative, so what i'm really doing to go back again is a partial integral.
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So taking that second equation, for example, i would say that f of x, y, is the partial integral of the partial derivative of f with respect to y.
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And because partial f partial y is q, this is just the same thing as the partial integral of q with respect to y, and we know what q is.
03:00
Q is x times the quantity xy plus 1.
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So i'm just doing the partial integral of x times xy plus 1 with respect to y.
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Now what in the heck is this partial integral about? what i really mean here is just like with a partial derivative i'm treating the other variable as a constant.
03:17
So i'm integrating with respect to y treating x as a constant and the other part of it is after i do the integration instead of an arbitrary constant in the end a plus c i'm going to get a plus i'm going to get a plus arbitrary function of the variable i was treating as a constant.
03:34
So when i'm doing this integral, i'm treating x as a constant that i can pull out.
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This is the integral then of x times xy plus 1 partial y, which is the same thing as x times the quantity, 1 half x, y squared plus y.
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Remember i'm integrating everything with respect to y, treating x as some constant like it was pi or whatever.
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And instead of a plus c, i'm going to tack on an arbitrary function, of my variable i was treating as a constant.
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And this is how we do partial integration.
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So what this tells me is the function i'm looking for is 1 half x squared y squared plus x y plus some function h of y.
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I'm not quite sure what is yet.
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So i haven't completely found the function because i don't know what this arbitrary function of h might be.
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Lucky for me, i was working with two equations and i've only used one of them so far.
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So to figure out what h is, i'm now going to leverage the second equation.
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I'm not going to do partial integration this time because i already have an expression for f.
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It makes more sense to just do partial differentiation.
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So taking the partial derivative of the function f that i found with respect to x, i get partial f, partial x equals xy squared plus y plus h prime...