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Model equation resolution

- What I'd like to do in this video is start exploring how we can model things with the differential equations. And in this video in particular, we will explore modeling population. Modeling population. We're actually going to go into some depth on this eventually, but here we're going to start with simpler models. And we'll see, we will stumble on using the logic of differential equations. Things that you might have seen in your algebra or your precalculus class. So, in some level, what we're going to do here is going to be review, but we're going to get there using the power of modeling with differential equations. So let's just define some variables. Let's say that P is equal to our population. And let's say that t is, let's say that t is equal to the time that has passed in days. In days, it could have been years or months. But let's say we're doing the population of insects that reproduce quite quickly. So days seem like a nice time span to care about. Now, what would be a reasonable model? Well, we could say that the rate of change, the rate of change of our population, with respect to time, with respect to time is, well, a reasonable thing to say is that it's going to be proportional to the actual population, the actual population. Why is that reasonable? Well, the larger the population, the larger the rate at any given time. If you have a thousand people, the rate at which they're reproducing is going to be more, or a thousand insects is going to be more insects per second, per day or per year than if you only have 10 insects. So it makes sense that the rate of growth of your population, with respect to time, is going to be proportional to your population. And so, you know, sometimes you think of differential equations as these daunting complex things, but notice we've just been able to express a reasonably not so complicated idea. The rate of change of population is going to be proportional to the population. And now, once we've expressed that, we can actually try to solve this differential equation. Find a general solution, and then we can try to put some initial conditions on there or some states of the population that we know to actually solve for the constants to find a particular solution. So how do we do that? And I encourage you to pause the video at any time and see if you can solve this differential equation. So assuming you, at least, maybe have had an attempt at it, and you might immediately recognize that this is a separable differential equation. And in separable differential equations, we want one variable and all the differentials involving that variable on one side, and the other variable and all the differentials involving the other variable on the other side, and then we can integrate both sides. And once again,...