Thurs. 10/30: Intro. to differential equations, solutions to differential equations, Euler’s method

10.1 and 10.2

1) Main points

10.1 introduces the concept of differential equation. A differential equation is an equation of the rate of change of a function, often we don't know the actual function we only know information that comprised the differential equation. This chapter also introduces a logistical differential equation which is a differential equation that explains a graph the satisfies the conditions of a logistic growth curve. 10.2 introduces how to 'solve' a differential equation which means to find an equation that works with the differential equation. To find if a function is the solution to a differential equation you: set the left side of the equation (the derivative of the function you are testing) = to the differential with the equation being tested substituted in as the variable. Because this new equation often involves a constant the determined function may be a family of possibly different functions because the constant is not defined. To find a specific function that works (by defining the constant) you substitute in a y value that you know. This exact function is called the "particular solution".


2) Challenges

It is really easy to mix up these equations when you are checking to see if they work together. It will be important to write and label the equation and function carefully. These problems require a good knowledge of derivates and the chain rule so it is important to differentiate everything carefully.


3) Reflections

Differential equations have a lot of real world applications especially because often you know the rate of change but not the original function. What we have learned so far about differential equations allows us to check whether a rate of change and a proposed equation correlate, this makes our real life data collection more reliable.