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Welcome back.

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I'm just trying to show you as many examples as possible of

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solving exact differential equations.

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One, trying to figure out whether the

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equations are exact.

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And then if you know they're exact, how do you figure out

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the psi and figure out the solution of the

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differential equation?

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So the next one in my book is 3x squared minus 2xy plus 2

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times dx, plus 6y squared minus x squared plus 3 times

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dy is equal to 0.

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So just the way it was written, this isn't

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superficially in that form that we want, right?

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What's the form that we want?

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We want some function of x and y plus another function of x

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and y, times y prime, or dy dx, is equal to 0.

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We're close.

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How could we get this equation into this form?

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We just divide both sides of this equation by dx, right?

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And then we get 3x squared minus 2xy plus 2.

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We're dividing by dx, so that dx just becomes a 1.

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Plus 6y squared minus x squared plus 3.

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And then we're dividing by dx, so that becomes dy dx, is

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equal to-- what's 0 divided by dx?

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Well it's just 0.

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And there we have it.

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We have written this in the form that we

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need, in this form.

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And now we need to prove to ourselves that this is an

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exact equation.

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So let's do that.

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So what's the partial of M?

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This is the M function, right?

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This was a plus here.

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What's the partial of this with respect to y?

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This would be 0.

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This would be minus 2x, and then just a 2.

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So the partial of this with respect to y is minus 2x.

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What's the partial of N with respect to x?

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This would be 0, this would be minus 2x.

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So there you have it.

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The partial of M with respect to y is equal to the partial

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of N with respect to x.

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My is equal to Nx.

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So we are dealing with an exact equation.

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So now we have to find psi.

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The partial of psi with respect to x is equal to M,

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which is equal to 3x squared minus 2xy plus 2.

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Take the anti-derivative with respect to x on both sides,

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and you get psi is equal to x to the third minus x squared

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y-- because y is just a constant-- plus 2x, plus some

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function of y.

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Right?

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Because we know psi is a function of x and y.

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So when you take a derivative, when you take a partial with

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respect to just x, a pure function of just y would get

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lost. So it's like the constant, when we first

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learned taking anti-derivatives.

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And now, to figure out psi, we just have to solve for h of y.

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And how do we do that?

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Well let's take the partial of psi with respect to y.

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That's going to be equal to this right here.

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So The partial of psi with respect to y, this is 0, this

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is minus x squared.

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So it's minus x squared-- this is o-- plus h prime of y, is

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going to be able to what?

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That's going to be equal to our n of x, y.

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It's going to be able to this.

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And then we can solve for this.

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So that's going to be equal to 6y squared minus x

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squared plus 3.

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You can add x squared to both sides to get

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rid of this and this.

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And then we're left with h prime of y is equal to 6y

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squared plus 3.

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Anti-derivative-- so h of y is equal to what is this-- 2y

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cubed plus 3y.

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And you could put a plus c there, but the plus c merges

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later on when we solve the differential equation, so you

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don't have to worry about it too much.

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So what is our function psi?

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I'll write it in a new color.

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Our function psi as a function of x and y is equal to x to

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the third minus x squared y plus 2x.

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Plus h of y, which we just solved for.

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So h of y is plus 2y to the third plus 3y.

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And then they're could be a plus c there, but you'll see

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that it doesn't matter much.

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Actually I want to do something

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a little bit different.

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I'm not just going to chug through the problem.

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I want to kind of go back to the intuition.

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Because I don't want this to be completely mechanical.

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Let me just show you what the derivative-- using what we

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knew before you even learned anything about the partial

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derivative chain rule-- what is the derivative of psi with

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respect to x.

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What is the derivative of psi with respect to x?

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Here we just use our implicit differentiation skills.

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So the derivative of this-- I'll do it in a new color-- 3x

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squared minus-- now we're going to have to use the chain

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rule here-- so the derivative of the first expression with

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respect to x is-- well, let me just put the minus sign and I

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could put like that-- so it's 2x times y plus the first

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function, x squared times the derivative of the second

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function with respect to x.

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Well that's just y prime, right?

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It's the derivative of y with respect to y is 1, times the

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derivative of y with respect to x, which is just y prime.

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Fair enough.

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Plus the derivative of this with respect to x is easy, 2.

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Plus the derivative of this with respect to x.

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Well let's take the derivative of this with respect to y

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first. We're just doing implicit differentiation of

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the chain rule.

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So this is plus 6y squared.

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And then we're using the chain rule, so we took the

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derivative with respect to y.

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And then you have to multiply that times the derivative of y

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with respect x, which is just y prime.

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Plus the derivative of this with respect to why is 3

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times-- we're just doing the chain rule-- the derivative of

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y with respect to x.

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So that's y prime.

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Let's try to see if we can simplify this.

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So we get this is equal to 3x squared minus 2xy plus 2.

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So that's this term, this term, and this term.

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Plus-- let's just put the y prime outside-- y prime

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times-- let's see, you have a negative sign out here-- minus

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x squared plus 6y squared plus 3.

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So this is the derivative of our psi as we solved it.

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Look at this closely and notice that that is the same--

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hopefully it's the same-- as our original problem.

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What was our original problem that we started working with?

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The original problem was 3x squared minus 2xy plus 2, plus

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6y squared minus x square plus 3, times y

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prime, is equal to 0.

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So this was our original problem.

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And notice that the derivative of psi with respect to x just

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using implicit differentiation is exactly this.

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So hopefully this gives you a little intuition of why we can

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just rewrite this equation as the derivative with respect x

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of psi, which is a function of x and y, is equal to 0.

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Because this is the derivative of psi with respect to x.

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I wrote out here.

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It's the same thing-- this right here-- right?

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So that equals 0.

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So if we take the anti-derivative of both sides,

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we know that the solution of this differential equation is

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that psi of x and y is equal to c as the solution.

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And we know what psi is, so we just set that equal to c, and

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we have the implicit-- we have a solution to the differential

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equation, I'll just define implicitly.

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So the solution-- you don't have to do this every time.

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This step right here you wouldn't have to do if you're

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taking a test, unless the teacher

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explicitly asked for it.

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I just wanted kind of make sure that you know what you're

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doing, that you're not just doing things completely

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mechanically.

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That you really see that the derivative of psi really does

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give you-- we solved for psi.

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And I just wanted to show you that the derivative of psi

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with respect to x, just using implicit differentiation and

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our standard chain rule, actually gives you the left

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hand side of the differential equation, which was our

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version of problem.

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And then that's how we know that that the derivative of

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psi with respect x is equal to 0, because our original

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differential equation was equal to 0.

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You take the anti-derivative of both sides of this, you get

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psi is equal to C, is the solution of the

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differential equation.

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Or if you wanted to write it out, psi is this thing.

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Our solution to the differential equation is x to

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the third, minus x squared y, plus 2x, plus 2y to the third,

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plus 3y, is equal to c, is the implicitly defined solution of

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our original differential equation.

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Anyway I've run out of time again.

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I will see you in the next video.

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