Exact Equations Example 3 One more exact equation example
Exact Equations Example 3
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- Welcome back.
- I'm just trying to show you as many examples as possible of
- solving exact differential equations.
- One, trying to figure out whether the
- equations are exact.
- And then if you know they're exact, how do you figure out
- the psi and figure out the solution of the
- differential equation?
- So the next one in my book is 3x squared minus 2xy plus 2
- times dx, plus 6y squared minus x squared plus 3 times
- dy is equal to 0.
- So just the way it was written, this isn't
- superficially in that form that we want, right?
- What's the form that we want?
- We want some function of x and y plus another function of x
- and y, times y prime, or dy dx, is equal to 0.
- We're close.
- How could we get this equation into this form?
- We just divide both sides of this equation by dx, right?
- And then we get 3x squared minus 2xy plus 2.
- We're dividing by dx, so that dx just becomes a 1.
- Plus 6y squared minus x squared plus 3.
- And then we're dividing by dx, so that becomes dy dx, is
- equal to-- what's 0 divided by dx?
- Well it's just 0.
- And there we have it.
- We have written this in the form that we
- need, in this form.
- And now we need to prove to ourselves that this is an
- exact equation.
- So let's do that.
- So what's the partial of M?
- This is the M function, right?
- This was a plus here.
- What's the partial of this with respect to y?
- This would be 0.
- This would be minus 2x, and then just a 2.
- So the partial of this with respect to y is minus 2x.
- What's the partial of N with respect to x?
- This would be 0, this would be minus 2x.
- So there you have it.
- The partial of M with respect to y is equal to the partial
- of N with respect to x.
- My is equal to Nx.
- So we are dealing with an exact equation.
- So now we have to find psi.
- The partial of psi with respect to x is equal to M,
- which is equal to 3x squared minus 2xy plus 2.
- Take the anti-derivative with respect to x on both sides,
- and you get psi is equal to x to the third minus x squared
- y-- because y is just a constant-- plus 2x, plus some
- function of y.
- Because we know psi is a function of x and y.
- So when you take a derivative, when you take a partial with
- respect to just x, a pure function of just y would get
- lost. So it's like the constant, when we first
- learned taking anti-derivatives.
- And now, to figure out psi, we just have to solve for h of y.
- And how do we do that?
- Well let's take the partial of psi with respect to y.
- That's going to be equal to this right here.
- So The partial of psi with respect to y, this is 0, this
- is minus x squared.
- So it's minus x squared-- this is o-- plus h prime of y, is
- going to be able to what?
- That's going to be equal to our n of x, y.
- It's going to be able to this.
- And then we can solve for this.
- So that's going to be equal to 6y squared minus x
- squared plus 3.
- You can add x squared to both sides to get
- rid of this and this.
- And then we're left with h prime of y is equal to 6y
- squared plus 3.
- Anti-derivative-- so h of y is equal to what is this-- 2y
- cubed plus 3y.
- And you could put a plus c there, but the plus c merges
- later on when we solve the differential equation, so you
- don't have to worry about it too much.
- So what is our function psi?
- I'll write it in a new color.
- Our function psi as a function of x and y is equal to x to
- the third minus x squared y plus 2x.
- Plus h of y, which we just solved for.
- So h of y is plus 2y to the third plus 3y.
- And then they're could be a plus c there, but you'll see
- that it doesn't matter much.
- Actually I want to do something
- a little bit different.
- I'm not just going to chug through the problem.
- I want to kind of go back to the intuition.
- Because I don't want this to be completely mechanical.
- Let me just show you what the derivative-- using what we
- knew before you even learned anything about the partial
- derivative chain rule-- what is the derivative of psi with
- respect to x.
- What is the derivative of psi with respect to x?
- Here we just use our implicit differentiation skills.
- So the derivative of this-- I'll do it in a new color-- 3x
- squared minus-- now we're going to have to use the chain
- rule here-- so the derivative of the first expression with
- respect to x is-- well, let me just put the minus sign and I
- could put like that-- so it's 2x times y plus the first
- function, x squared times the derivative of the second
- function with respect to x.
- Well that's just y prime, right?
- It's the derivative of y with respect to y is 1, times the
- derivative of y with respect to x, which is just y prime.
- Fair enough.
- Plus the derivative of this with respect to x is easy, 2.
- Plus the derivative of this with respect to x.
- Well let's take the derivative of this with respect to y
- first. We're just doing implicit differentiation of
- the chain rule.
- So this is plus 6y squared.
- And then we're using the chain rule, so we took the
- derivative with respect to y.
- And then you have to multiply that times the derivative of y
- with respect x, which is just y prime.
- Plus the derivative of this with respect to why is 3
- times-- we're just doing the chain rule-- the derivative of
- y with respect to x.
- So that's y prime.
- Let's try to see if we can simplify this.
- So we get this is equal to 3x squared minus 2xy plus 2.
- So that's this term, this term, and this term.
- Plus-- let's just put the y prime outside-- y prime
- times-- let's see, you have a negative sign out here-- minus
- x squared plus 6y squared plus 3.
- So this is the derivative of our psi as we solved it.
- Look at this closely and notice that that is the same--
- hopefully it's the same-- as our original problem.
- What was our original problem that we started working with?
- The original problem was 3x squared minus 2xy plus 2, plus
- 6y squared minus x square plus 3, times y
- prime, is equal to 0.
- So this was our original problem.
- And notice that the derivative of psi with respect to x just
- using implicit differentiation is exactly this.
- So hopefully this gives you a little intuition of why we can
- just rewrite this equation as the derivative with respect x
- of psi, which is a function of x and y, is equal to 0.
- Because this is the derivative of psi with respect to x.
- I wrote out here.
- It's the same thing-- this right here-- right?
- So that equals 0.
- So if we take the anti-derivative of both sides,
- we know that the solution of this differential equation is
- that psi of x and y is equal to c as the solution.
- And we know what psi is, so we just set that equal to c, and
- we have the implicit-- we have a solution to the differential
- equation, I'll just define implicitly.
- So the solution-- you don't have to do this every time.
- This step right here you wouldn't have to do if you're
- taking a test, unless the teacher
- explicitly asked for it.
- I just wanted kind of make sure that you know what you're
- doing, that you're not just doing things completely
- That you really see that the derivative of psi really does
- give you-- we solved for psi.
- And I just wanted to show you that the derivative of psi
- with respect to x, just using implicit differentiation and
- our standard chain rule, actually gives you the left
- hand side of the differential equation, which was our
- version of problem.
- And then that's how we know that that the derivative of
- psi with respect x is equal to 0, because our original
- differential equation was equal to 0.
- You take the anti-derivative of both sides of this, you get
- psi is equal to C, is the solution of the
- differential equation.
- Or if you wanted to write it out, psi is this thing.
- Our solution to the differential equation is x to
- the third, minus x squared y, plus 2x, plus 2y to the third,
- plus 3y, is equal to c, is the implicitly defined solution of
- our original differential equation.
- Anyway I've run out of time again.
- I will see you in the next video.
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