00:01
So here we are trying to find an expression that tracks the population.
00:07
And we're assuming that the population y in millions satisfies this differential equation, d -y -t equals k -y.
00:17
And then we're assuming that we know a couple of things about the population, that at times zero, which corresponds to the year 2000, is 35 .6 million.
00:27
And at t equals 50, the year 2050 is 102 .6 million.
00:35
So first we can just find the general form of the solution to the equation, and then we'll use these two points on the curve to solve for the constants that we don't know.
00:46
This equation is separable, so we can just move this y term over to the left, and the d t over to the right, and integrate both sides.
01:02
On the left over here, we'll get natural log of the absolute value of y.
01:07
But in real world context, y is only positive, right, because it's tracking the population.
01:13
So we can just drop the absolute value eventually.
01:16
Then on the right, we'll get kt, and then plus some constant.
01:22
Remember, we're grouping together, the constant on the right and the left.
01:25
And so then if we take both sides to the e, then we can bring down this constant.
01:36
And we'll get like e to the kt plus c, which is e to the kt times e to the c.
01:44
We can rewrite e to the c as just some constant d, and we get that this is the general form of our equation y...