Exact equations and integrating factors
Exact Equations Example 2 Some more exact equation examples
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- Let's do some more examples with exact
- differential equations.
- And I'm getting these problems from page 80 of my old college
- differential equations books.
- This is the fifth edition of Elementary Differential
- Equations by William Boyce and Richard DiPrima.
- I want to make sure they get credit, that I'm not making up
- these problems. I'm getting it from their book.
- Anyway, so I'm just going to give a bunch of equations.
- We have to figure out if they're exact, and if they are
- exact, we'll use what we know about exact differential
- equations to figure out their solutions.
- So the first one they have is, 2x plus 3, plus 2y minus 2,
- times y prime is equal to 0.
- So this is our M of x and y-- although, this is only a
- function of x-- and then this is our N, right?
- You could say that's M, or that's N.
- You could also say that, if this is exact-- well, first
- let's [? test ?]
- this exact, before we start talking about psi.
- So what's the partial of this, with respect to y?
- The partial of M, with respect to y.
- Well, there's no y here, so it's 0.
- The rate of change that this changes with
- respect to y is 0.
- And what's the rate of change this changes,
- with respect to x?
- The partial of N, with respect to x is equal to-- well,
- there's no x here, right?
- So these are just constants from an x point of view, so
- this is all going to be 0.
- But we do see that they're both 0.
- So M sub y, or the partial with respect to y, is equal to
- the partial with respect to x.
- So this is exact.
- And actually, we don't have to use exact equations here.
- We'll do it, just so that we get used to it.
- But if you look here, you actually could have figured
- out that this is actually a separable equation.
- But anyway, this is exact.
- So knowing that it's exact, it tells us that there's some
- function psi, where psi is a function of x and y.
- Where psi sub x is equal to this function, is equal to 2x
- plus 3, and psi-- I shouldn't say sub x.
- I say the partial of psi, with respect to x.
- And the partial of psi, with respect to y, is equal to
- this, 2y minus 2.
- And if we can find our psi, we know that this is just the
- derivative of psi.
- Because we know that the derivative, with respect to x
- of psi, is equal to the partial of psi, with respect
- to x, plus the partial of psi, with respect
- to y, times y prime.
- So this is this just the same form as that.
- So if we can figure out y, then we can rewrite this
- equation as dx, the derivative of psi, with respect to x, is
- equal to 0.
- Let me switch colors, or it's going to get monotonous.
- This right here, if we can find a psi, where the partial
- with respect to x, is this, the partial with respect to y,
- is this, then this can be rewritten as this.
- And how do we know that?
- Because the derivative of psi, with respect to x, using the
- partial derivative chain rules, is this.
- This partial with respect to x, that's this.
- This partial with respect to y, is this, times y prime.
- So this is the whole point of exact equations.
- But anyway, so let's figure out what our psi is.
- Actually, before we figure out, if the derivative of psi,
- with respect to x, is 0, then if you integrate both sides,
- you just-- the solution of this equation is
- psi is equal to c.
- So using this information, if we can solve for psi, then we
- know that the solution of this differential equation is psi
- is equal to c.
- And then if we have some initial conditions, we could
- solve for c.
- So let's solve for psi.
- So let's integrate both sides of this equation,
- with respect to x.
- And then we get psi is equal to x squared plus 3x, plus
- some function of y.
- Let's call it h of y.
- And remember, normally when you take an antiderivative,
- you have just a plus c here, right?
- But you can kind of say we took an anti-partial
- derivative.
- So when you took a partial derivative, with respect to x,
- not only do you lose constants-- that's why we have
- a plus c, normally-- but you also lose anything that's a
- function of just y, and not x.
- So for example, take the partial derivative of this
- with respect to x, you're going to get this, right?
- Because the partial derivative of a function, purely of y,
- with respect to x, is going to be 0.
- So it will disappear.
- So anyway, we take the antiderivative of
- this, we get this.
- Now, we use this information.
- Well, we use this information.
- We take the partial of this expression, and we say, well,
- the partial of this expression, with respect to y,
- has to equal this, and then we can solve for h of y, then
- we'll be done.
- So let's do that.
- So the partial of psi, with respect to y, is equal to--
- well, that's going to be 0, 0, 0.
- This part is a function of x.
- If you take the partial with respect to y, it's 0, because
- these are constants, from a y point of view.
- So you're left with h prime of y.
- So we know that h prime of y, which is the partial of psi,
- with respect to y, is equal to this.
- So h prime of y is equal to 2y minus 2.
- And then if we wanted to figure out what h of y is, we
- get h of y-- just integrate both sides, with respect to
- y-- is equal to y squared plus-- sorry-- y
- squared minus 2y.
- Now, you could have a plus c there, but if you watched the
- previous example, you'll see that that c kind of merges
- with the other c, so you don't have to worry
- about it right now.
- So what is our psi function, as we know it now, not
- worrying about the plus c?
- It is psi of x and y is equal to x squared plus 3x, plus h
- of y-- which we figured out is this-- plus y
- squared, minus 2y.
- And we know a solution of our original differential equation
- is psi is equal to c.
- So the solution of our differential equation is this
- is equal to c.
- x squared plus 3x, plus y squared, minus
- 2y is equal to c.
- If you had some additional conditions, you could test it.
- And I encourage you to test this out on this original
- equation, or I encourage you to take the derivative of psi,
- and prove to yourself that if you took the derivative of
- psi, with respect to x, here, implicitly, that you would get
- this differential equation.
- Anyway, let's do another one.
- Let's clear image.
- So the more examples you see, the better.
- So let's see, this one says 2x plus 4y, plus 2x minus 2y, y
- prime is equal to 0.
- So what's the partial of this with respect to y?
- So M, the partial of M, with respect to y-- this is 0-- so
- it's equal to 4.
- What's the partial of this, with respect to x, just this
- part right here?
- The partial of N, with respect to x, is 2.
- This is 0.
- So the partial of this, with respect to y, is different
- than the partial of N, with respect to x.
- So this is not exact.
- So we can't solve this using our exact methodology.
- So that was a fairly straightforward problem.
- Let's do another one.
- Let's see.
- I'm running out of time, so I want to do one that's not too
- complicated.
- Let's see, 3x squared minus 2xy-- actually, let me do this
- in the next problem.
- I don't want to rush these things.
- I will continue this in the next video.
- See you soon.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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