- [Tutor] What we're going to do in this video
is see if we can solve the differential equation,
the derivative of y with respect to x
is equal to x times y,
pause this video
and see if you can find a general solution here.
So the first thing that my brain likes to do,
when I see a differential equation is to say,
hey, is this separable and when I say separable,
can I get all the ys and dys on one side
and all the xs and dxs on the other side?
And you can indeed do that,
if we treat our differentials like,
if we could treat 'em like algebraic variables,
which is fair game, when you're dealing
with differential equations like this,
we could multiple both sides by dx,
so multiple both sides by dx
and we could divide both sides by y,
let me move this over a little bit, so we have some space,
so we could also multiple both sides by one over y,
one over y and so what that does is
these dxs cancel out and this y and one over y cancels out
and we are left with, let me write all the things
in terms of y on the left-hand side in blue,
so we have one over you
dy is equal to,
and then I'll do all this stuff in orange,
we have is equal to, we're just left
with an x and a dx, x dx
and then we'll wanna take
the indefinite integral of both sides,
now what's the anti-derivative of one over y?
Well, if we want it in the most general form,
this would be the natural log of the absolute value of y
and then this is going to be equal to
the anti-derivative of x
is x squared over two
and then we wanna put a constant on either side,
I'll just put it on the right-hand side, plus C,
this ensures that we're dealing with the general solution,
now if we wanna solve explicitly for y,
we could raise e to both sides' power,
another way to think about it is,
if this is equal to that,
then e to this power
is going to be the same thing as e to that power,
now, what does that do for us?
Well, what is e to the natural log
of the absolute value of y?
Well, I'm raising e to the power
that I would have to raise e too
to to get to the absolute value of y,
so the left-hand side here,
it simplifies to the absolute value of y
and we get that as being equal to,
now, we could use our exponent properties,
this over here is the same thing as
e to the x squared over two
times e to the C times e to the C,
I am just using our exponent properties here,
well, e to the C, we could just view that
as some other type of constant, this is just sum constant C,
so we could rewrite this whole thing as
C e to the x squared
x squared over two, hopefully you see what I'm doing there,
I just use my exponent properties
e dah dah dah, sum of two things
is equal to e to the first thing times e to the second thing
and I just said, well, e to the power of sum constant C,
we could just relabel that as, let's call that our blue C
and so this simplifies to blue C times e
to the x squared over two,
now we saw this absolute value sign here,
so this essentially means that y
could be equal to, we could write it this way,
y could be equal to plus or minus C,
C e to the x squared over two,
x squared over two,
but once again, we don't know what this constant is,
I didn't say that it was positive or negative,
so when you see a plus or minus of a constant here,
you could really just subsume all of this,
I'll just call this red C,
so we could say that y is equal to,
I'll just rewrite it over again for fun,
y is equal to red C,
not the Red Sea, but a red C (laughs)
times e to the x squared over two,
this right over here, it's the general solution
to the original, separable differential equation.