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OK, I filled your brain with a bunch of partial derivatives

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and psi's, with respect to x's and y's.

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I think now it's time to actually do it with a real

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differential equation, and make things a

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little bit more concrete.

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So let's say I have the differential, y, the

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differential equation, y cosine of x, plus 2xe to the

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y, plus sine of x, plus-- I'm already running out of space--

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x squared, e to the y, minus 1, times y

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prime, is equal to 0.

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Well, your brain is already, hopefully, in exact

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differential equations mode.

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But if you were to see this pattern in general, where you

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see a function of x and y, here-- this is just some

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function of x and y-- and then you have another function of x

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and y, times y prime, or times dy, d of x, your brain should

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immediately say if this is inseparable.

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And I'm not going to try to make it separable, just

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because that'll take a lot of time.

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But if it's not separable, your brain said, oh, maybe

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this is an exact equation.

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And, you say, let me test whether

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this is an exact equation.

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So if this is an exact equation, this is our function

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M, which is a function of x and y.

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And this is our function N, which is a

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function of x and y.

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Now, the test is to see if the partial of this, with respect

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to y, is equal to the partial of this, with respect to x.

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So let's see.

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The partial of M, with respect to y, is equal to-- let's see,

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y is-- so this cosine of x is just a constant, so it's just

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cosine of x.

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Cosine of x plus-- now, what's the derivative?

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Well, 2x is just a constant, what's the derivative of e to

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the y, with respect to y?

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Well, it's just e to the y, right?

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So we have the constant on the outside, 2x times the

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derivative, with respect to y, so it's 2xe to the y.

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Fair enough.

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Now, what is the partial derivative of this, with

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respect to x?

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So N sub x, or the partial of N, with respect to x-- so

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what's the derivative of sine of x, with respect to x?

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Well, that's easy, that's cosine of x, plus 2x times e

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to the y, right? e and y is just a constant, because y is

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constant when we're taking the partial, with respect to x.

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So plus 2xe to the y.

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And then minus 1, the derivative of a constant, with

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respect to anything is going to be 0.

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So the derivative of N-- the partial of N, with respect to

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x, is cosine of x, plus 2xe to the y, which, lo and behold,

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is the same thing as the derivative, the partial of M,

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with respect to y.

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So there we have it.

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We've shown that M of y is equal to-- or the partial of

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M, with respect to y-- is equal to the partial of N,

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with respect to x, which tells us that

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this is an exact equation.

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Now, given that this is an exact equation-- oh, my wife

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snuck up behind me, I was wondering.

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I thought there was some critter in

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my house, or something.

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Anyway, so we know that this is an exact equation, so what

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does that tell us?

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Well, that tells us that there's some psi, where the

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partial derivative of psi, with respect to x, is equal to

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M, and the partial derivative of psi, with respect to y, is

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equal to N.

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And if we know that psi, then we can rewrite our

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differential equation as the derivative of psi, with

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respect to x, is equal to 0.

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So let's solve for psi.

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So we know that the partial of psi, with respect to x, is

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equal to M.

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So we could write that.

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We could write the partial of psi, with respect to x, is

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equal to M, which is y cosine of x, plus 2xe to the y.

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That's just here.

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That's my M of x.

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We could have done it the other way.

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We could have said the partial of y-- the partial of psi,

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with respect to y, is this thing over here.

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But let's just do it with x.

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Now, to at least get kind of a first approximation of what

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size-- not an approximation, but to start to get a sense of

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it-- let's take the derivative of both sides, with respect

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to-- sorry, take the antiderivative-- take the

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integral of both sides, with respect to x.

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So if you take the derivative of this, with respect to x, if

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you integrate-- sorry, if you were to take the

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antiderivative of this, with respect to x.

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So let me just write that down.

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The partial, with respect to x.

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We're going to [? integrate ?]

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it, with respect to x.

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That is going to be equal to the integral of this whole

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thing, with respect to x.

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Cosine of x plus 2xe to the y.

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We're integrating with respect to x.

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And normally when you integrate with respect to x,

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you'd say, OK, plus c, right?

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But it actually could be a plus-- since this is a

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partial, with respect to x, we could have had some function

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of y here in general, because y, we treat it

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as a constant, right?

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And that makes sense, because if you were to take the

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partial of both sides of this, with respect to x, if you were

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to take the partial of a function that is only a

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function of y, with respect to x, you would

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have gotten a 0 here.

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So when you take the antiderivative, you were,

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like, oh well, there might have been some function of y

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here that we lost when we took the partial,

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with respect to x.

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So anyway, this will simplify to psi.

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psi is going to be equal to the integral, with respect to

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x, or the antiderivative, with respect to x, here, plus some

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function of y that we might have lost when we took the

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partial, with respect to x.

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So let's do that.

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Let's figure out this integral.

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I'll do it in blue So y is just a constant.

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So the antiderivative of y cosine of x, is just y sine of

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x, plus-- either the y is constant, so 2x.

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The antiderivative of 2x, with respect to x, is x squared, so

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it's x squared e to the y.

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And then plus some function of y.

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And if you want to verify this, you should take the

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partial of this, with respect to x.

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If you take the partial of this, with respect to x,

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you're going to get this in here, which is our function,

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M, up here.

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And then when you take the partial of this, with respect

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to x, you'll get 0, and it'll get lost. OK, so

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we're almost there.

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We've almost figured out our psi, but we still need to

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figure out this function of y.

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Well, we know that if we take the partial of this, with

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respect to y, since this is an exact equation,

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we should get this.

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We should get our N function.

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So let's do that.

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So the partial-- I'll switch notation, just to expose you

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to it-- the partial psi, with respect to y, is going to be

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equal to-- so here, y sine of x, sine of x is just a

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constant. y is just y, so the derivative of this, with

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respect to y, is just sine of x.

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Plus the derivative of e to the y is e to the y. x squared

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is just a constant.

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So it's just x squared e to the y, plus-- what's the

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partial of f of y, with respect to y?

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It's going to be f prime of y.

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Well, what did we do?

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We took M, we integrated with respect to x, and we said,

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well, we might have lost some function of y, so we added

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that to it.

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And then we took the partial of that side that we've almost

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constructed, and we took the partial of that,

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with respect to y.

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Now, we know, since this is exact, that that is going to

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equal our N.

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So our N is up there.

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Cosine of x plus-- So that's going to be equal to-- I want

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to make sure I can read it up there-- to our N, right?

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Oh no, sorry.

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N is up here.

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Our N is up here.

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Sine of x-- let me write that-- sine of x plus x

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squared, e to the y, minus 1.

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So sine of x plus x squared, e to the y, minus 1.

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That was just our N, from our original

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differential equation.

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And now we can solve for f prime of y.

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So let's see, we get sine of x plus x squared, e to the y,

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plus f prime of y, is equal to sine of x plus x squared, e to

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the y, minus 1.

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So let's see, we can delete sine of x from both sides.

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We can delete x squared e to the y from both sides.

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And then what are we left with?

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We're left with f prime of y is equal to 1.

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And then we're left with f of y is equal to-- well, it

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equals y plus some constant, c, right?

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So what is our psi now?

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We wrote our psi up here, and we had this f of y here, so we

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can rewrite it now.

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So psi is a function of x and y-- we're actually pretty much

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almost done solving it-- psi is a function of x and y is

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equal to y sine of x, plus x squared, e to the y, plus y--

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oh, sorry, this is f prime of y, minus 1.

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So this is a minus 1.

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So this is a minus y plus c.

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So this is going to be a minus y plus c.

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So we've solved for psi.

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And so what does that tell us?

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Well, we said that original differential equation, up

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here, using the partial derivative chain rule, that

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original differential equation, can be rewritten now

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as the derivative dx of psi is equal to-- psi is a function

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of x and y-- is equal to 0.

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Or if you were to integrate both sides of this, you would

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get that psi of xy is equal to c is a solution of that

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differential equation.

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So if we were to set this is equal to c, that's the

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differential equation.

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So we could say, y sine of x plus x squared, e to the y,

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minus y-- now we could say, plus this c-- plus this c, you

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call it c1, is equal to c2.

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Well, you could subtract the c's from both sides, and just

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be left with a c at the end.

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But anyway, we have solved this exact equation, one,

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first, by recognizing it was exact, by taking the partial

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of this, with respect to y, and seeing if that was equal

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00:11:01,669 --> 00:11:04,110
to the partial of N, with respect to x.

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Once we saw that they were equal, we're like, OK, this is

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00:11:07,000 --> 00:11:08,090
going to be exact.

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So let's figure out psi.

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Since this is exact, M is going to be the partial of

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psi, with respect to x.

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N is the partial of psi, with respect to y.

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Then to figure out y, we integrated M, with respect to

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x, and we got this.

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But since we said, oh, well, instead of a plus c, it could

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have been a function of y there, because we took the

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partial, with respect to x, so this might have been lost. To

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figure out the function of y, we then took our psi that we

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figured out, took the partial of that, with

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respect to y, got this.

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And we said, this was an exact equation, so this is going to

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equal our N of x y.

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We set those equal to each other, and then we

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solved for f of y.

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And then we had our final psi.

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Our final psi was this.

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And then the differential equation, because of the chain

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rule of partial derivatives, we could rewrite the

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differential equation as this.

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The solution is this, and so this is the solution to our

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differential equation.

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See you in the next video.

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