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