Posted: February 19th, 2023
Hi, Class. We’ve talked quite a bit in weeks 3 and 4 here (in particular in the Lectures and Engineering Applied Exercises) about how sometimes a mathematical model is not easily solved analytically or with software. You might wonder: where do these wildly complicated differential equations come from if they’re too complicated to solve? Well–just because an equation is difficult to solve, that doesn’t mean it was difficult to create. Often we are able to create a differential equation as the starting point, or the most intuitive mathematical description of what we observe in real world phenomenon. A good very simple example of this is the exponential growth model. We might have some familiarity with
and its appearance in things like the continuously compounded interest formula or in a basic population growth model–but familiarity isn’t the same as intuition or understanding. When we look at the differential equation, things actually make much MORE sense:
If P is population, t is time, and k is a local, observable growth rate:
Check out the equation:
This equation tell us that the population grows at a rate that is proportional to the size of the population.
That’s a pretty simple and intuitive thing to write down! However, it’s not until this differential equation is solved that it’s easily useable:
For an explanation of that solution, check out this video! https://www.youtube.com/watch?v=qPzTJeCEAiU
The video you shared provides a clear and concise explanation of how to solve the exponential growth model’s differential equation. The solution to this equation gives us a formula for the population as a function of time, which we can use to make predictions about future population sizes.
The exponential growth model is just one example of how differential equations can be used to model real-world phenomena. Many other natural phenomena can also be modeled using differential equations, such as the spread of diseases, the flow of fluids, and the behavior of electrical circuits.
While some differential equations can be solved analytically or with the help of software, others may require more advanced mathematical techniques, such as numerical methods or approximation methods. Nonetheless, even if a differential equation cannot be solved exactly, it can still provide valuable insights into the behavior of the system being modeled.
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