- [Narrator] Welcome to a lesson on determining the region
in the xy-plane for which a first order
differential equation would have a unique solution
through a point.
If we have a first order differential equation
in standard form, as we see here,
with this initial condition,
if f of x comma y
and the partial derivative of f with respects to y
are both continuous on a rectangular region
defined by a less than x, less than b,
and c less than y, less than d,
that contains the point x of zero, y sub zero
given by the initial condition,
then there exists an interval centered at x sub zero,
and a unique function y of x defined on the interval
that satisfies the initial value problem.
So, our main goal here is to find the region
where f and the partial derivative are both continuous.
Notice how this region depends on both x and y,
unlike the linear first order differential equations
that only depend on x.
Let's take a look at our first example.
We want to determine the region in the xy-plane
for which the given DE has a unique solution
through the given point.
The first step is to recognize that we do have
a first order differential equation,
but it's not written in the correct form.
We want our differential equation in this form here.
So we need to solve this for dy, dx,
or y-prime in this case.
So we're going to divide both sides by x.
So we'll have y-prime,
equals y divided by x.
Which means f of x comma y,
is equal to y divided by x.
And now we'll find the partial of f
with respects to y.
So, we'll treat x as a constant.
And since the derivative of y with respects to y is one,
we just have one over x, or one divided by x.
So now, if we look at f,
the only restriction here is that we know
x can't equal zero,
because we'd have division by zero.
Therefore, f will be continuous on the region
when x is less than zero,
or
when x is greater than zero.
Notice how these would just be two half-planes.
If we look at the partial derivative,
notice the restriction is the same,
x can't equal zero.
So, the partial derivative is continuous
over the same region when x is less than zero,
or when x is greater than zero.
Normally these two regions won't be the same,
so we'd be looking for the inner section
of these two regions.
But in this case, since they are the same,
the original differential equation has unique solutions
over the region when x is less than zero,
or when x is greater than zero.
Remember, these are regions in the xy-plane,
so these are two half-planes.
Let's take a look at another example.
Same question, different differential equation.
Notice again, it's not in the correct form.
So, we'll divide both sides by the quantity
one minus y squared, to solve for y-prime.
So we'd have y-prime equals x squared
divided by the quantity one minus y squared.
So, now we should recognize that f of x comma y
is equal to x squared divided by the quantity
one minus y squared.
Now, let's find the partial derivative
of f with respects to y.
It might help to think of f
as x squared times the quantity one minus y to the second,
to the negative one power.
Again, we'll treat x as a constant,
so we'll apply the chain rule here,
multiply by negative one.
So we'll have negative x squared
times the quantity one minus y squared
to the negative two times negative two y.
So, our partial derivative
is going to be positive two x squared y
all over the quantity one minus y squared, squared.
Looking at function f,
the only restriction here is we can't have a zero
in the denominator or division by zero.
So, we know that one minus y squared can't equal zero.
Or one can't equal y squared.
If we square root both sides.
So, we know y can't equal plus or minus one.
Everywhere else, f is going to be continuous.
And again, if we look at the partial derivative,
notice how the denominator is going to be zero,
again, when one minus y squared is equal to zero.
So, the partial derivative has the same restrictions.
Which means the differential equation
will have unique solutions over the entire xy-plane
except when y equals plus or minus one.
Which means y could be less than negative one,
or between negative one and one,
or y could be greater than one.
And this is the region that we're looking for.
I think we'll stop here and call this part one.
We'll look at two more examples in part two.
I hope this has been helpful.