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- I think it's reasonable to do one more separable
- differential equations problem, so let's do it.
- The derivative of y with respect to x is equal to y
- cosine of x divided by 1 plus 2y squared, and they give us
- an initial condition that y of 0 is equal to 1.
- Or when x is equal to 0, y is equal to 1.
- And I know we did a couple already, but another way to
- think about separable differential equations is
- really, all you're doing is implicit
- differentiation in reverse.
- Or another way to think about it is whenever you took an
- implicit derivative, the end product was a separable
- differential equation.
- And so, this hopefully forms a little bit of a connection.
- Anyway, let's just do this.
- We have to separate the y's from the x's.
- Let's multiply both sides times 1 plus 2y squared.
- We get 1 plus 2y squared times dy dx is equal
- to y cosine of x.
- We still haven't fully separated the y's and the x's.
- Let's divide both sides of this by y, and then let's see.
- We get 1 over y plus 2y squared divided by y, that's
- just 2y, times dy dx is equal to cosine of x.
- I can just multiply both sides by dx.
- 1 over y plus 2y times dy is equal to cosine of x dx.
- And now we can integrate both sides.
- So what's the integral of 1 over y with respect to y?
- I know your gut reaction is the natural log of y, which is
- correct, but there's actually a slightly broader function
- than that, whose derivative is actually 1 over y, and that's
- the natural log of the absolute value of y.
- And this is just a slightly broader function, because it's
- domain includes positive and negative numbers, it just
- excludes 0.
- While natural log of y only includes
- numbers larger than 0.
- So natural log of absolute value of y is nice, and it's
- actually true that at all points other than 0, its
- derivative is 1 over y.
- It's just a slightly broader function.
- So that's the antiderivative of 1 over y, and we proved
- that, or at least we proved that the derivative of natural
- log of y is 1 over y.
- Plus, what's the antiderivative of 2y with
- respect to y?
- Well, it's y squared, is equal to-- I'll do the
- plus c on this side.
- Whose derivative is cosine of x?
- Well, it's sine of x.
- And then we could add the plus c.
- We could add that plus c there.
- And what was our initial condition? y of
- 0 is equal to 1.
- So when x is equal to 0, y is equal to 1.
- So ln of the absolute value of 1 plus 1 squared is equal to
- sine of 0 plus c.
- The natural log of one, e to the what power is 1?
- Well, 0, plus 1 is-- sine of 0 is 0 --is equal to C.
- So we get c is equal to 1.
- So the solution to this differential equation up here
- is, I don't even have to rewrite it, we figured out c
- is equal to 1, so we can just scratch this out, and
- we could put a 1.
- The natural log of the absolute value of y plus y
- squared is equal to sine of x plus 1.
- And actually, if you were to graph this, you would see that
- y never actually dips below or even hits the x-axis.
- So you can actually get rid of that absolute
- value function there.
- But anyway, that's just a little technicality.
- But this is the implicit form of the solution to this
- differential equation.
- That makes sense, because the separable differential
- equations are really just
- implicit derivatives backwards.
- And in general, one thing that's kind of fun about
- differential equations, but kind of not as satisfying
- about differential equations, is it really is just a whole
- hodgepodge of tools to solve different types of equations.
- There isn't just one tool or one theory that will solve all
- differential equations.
- There are few that will solve a certain class of
- differential equations, but there's not just one
- consistent way to solve all of them.
- And even today, there are unsolved differential
- equations, where the only way that we know how to get
- solutions is using a computer numerically.
- And one day I'll do videos on that.
- And actually, you'll find that in most applications, that's
- what you end up doing anyway, because most differential
- equations you encounter in science or with any kind of
- science, whether it's economics, or physics, or
- engineering, that they often are unsolveable, because they
- might have a second or third derivative involved, and
- they're going to multiply.
- I mean, they're just going to be really complicated, very
- hard to solve analytically.
- And actually, you are going to solve them numerically, which
- is often much easier.
- But anyway, hopefully at this point you have a pretty good
- sense of separable equations.
- They're just implicit differentiation backwards, and
- it's really nothing new.
- Our next thing we'll learn is exact differential equations,
- and then we'll go off into more and more methods.
- And then hopefully, by the end of this playlist, you'll have
- a nice toolkit of all the different ways to solve at
- least the solvable differential equations.
- See you in the next video.