 In this video, suppose that the population p of a flock of birds is growing exponentially so that dp over dx is equal to 20e to the point 05x where x is measured in times of years. So we're saying that the derivative p prime is equal to an exponential function here. So before anyone panics here, right, we're not saying that the population of the birds is growing exponentially. We're saying the rate at which the population of the birds is growing is growing exponentially. The derivative is growing exponentially. Yousers, what does that tell us for the function? It turns out it's not going to be much more complicated. But we want to find p in terms of x. So we want to write p as a function of x if we know there are 20 birds at the beginning of our observations here. So we have an initial value problem placed in front of us, a differential equation. We know a relationship on how the population is growing over time. We have an initial observation and we want to find an equation to model the population at a given time, right? So we know that p prime is equal to 20, excuse me, 20e to the point 05x. Now since we know the derivative is a function of x, we can actually take the antiderivative here, take the integral of dp, take the integral of 20e to the point 05x dx. The left-hand side is just going to be p. The right-hand side, we're going to get 20 over point 05e to the point 05x plus a constant, like so. And here 20 divided by point 05, be aware that point 05 is actually one-twentieth if you divide by one-twentieth, you're times it by 20. We're going to get that the population equals 400, 20 times 20, e to the point 05x plus a constant here. This gives us the general solution. The general solution to this differential equation. We want to find a particular solution and we do that with the initial value given. So when they say there are 20 birds initially, what they're saying is that p of 0 initially means that the start x equals 0. Initially there were 20 birds, so p of 0 equals 20. So if we make those substitutions in, let's see, p is 20 and you get 400e to the point 05 times 0. It's always nice when the initial value is 0, it doesn't have to be, but arithmetic with 0 is generally pretty nice. So what we see here is you're going to get e to the 0 power, which of course is 1, 400 times 1 is 400. We have 400 plus c, so if you subtract 400 from both sides, you're going to get that c equals negative 380. And we'll see in the future that this number right here is negative 380, it's measuring essentially some initial conditions about the population growth. This is actually measured in some type of elbow room, like how much does, how much potential of growth is going on right now, but we'll come back to that some time in the future. But for now, we can mention that our population of the birds would equal 400e to the negative 0.05x minus 380. And so after x many years, you could use this equation to predict how many birds are there going to be in the population. And so this example, this solving this initial value problem is starting to show you why people are interested in differential equations, because if we are some biologists who are studying this flock of birds, we are going to be interested in their population size. Can we come up with a mathematical equation that can predict how many birds we'll have on any given year, right? We can use this as we're studying the birds, but how does one actually determine this population? Well, in this situation, we use the growth rate, like we know how quickly they're growing and using that we are able to construct this differential equation. And so that brings us to the end of lecture 25 here. In the next lecture, we're going to start to see some more advanced techniques of solving differential equations, particularly we'll talk about a technique called separation of variables, in which case it turns out this example and the previous example we are doing are just a special case of that separation of variables situation.