10.4 Exponential Growth and Decay & 10.5 Applications and Modeling

Main Points:
The general solution to the differential equation dy/dt = ky is y = Ce^kt (C is a constant). When k is greater than 0, the function is exponentially growing, if k is less than 0, the function is exponentially decaying. Population growth, pollution concentration, and interest are all situations that can be solved by this model. Another common model for a function used in a differential equation is dy/dt = k(y - A). In this form k and A are constants. The general solution for this differential equation is y = A+Ce^kt. Equilibrium solutions are ones that give a constant for all values of the independent variable. This yields a horizontal line. An equilibrium solution is stable if a small change in the initial conditions produces solution that approaches the equilibrium as the independent variable approaches infinity. An equilibrium solution is unstable if the new solution veers away from the equilibrium as the variable approaches infinity.

Challenges:
I think I have a decent grip on the subjects talked about in these chapters. The most challenging thing for me was following book's description of solving the differential equation dy/dt = k(y - A). Ce^kt justs pops in there unexpectedly, so it's a little confusing at first.

Reflections:
Well now I got to see how to find the differential equation, which is good since it was bothering me that we couldn't necessarily solve to find the equation. This new tactic is nice, but it only works in some situations. The situations are pretty broad, so it is usefull.