Last week we introduced differential equations. Today, we practiced working with differential equations. Our first example was the warm-up initial value problem:
After that, we did several applied examples, starting with Newton’s Law of Cooling: The rate of change in the temperature of an object is directly proportional to the difference between the temperature of the object and its surroundings.
Translated into an equation, this becomes: \[ \frac{dT}{dt} = -k (T - T_S)\] where \(T\) is the temperature of the object, \(t\) is the time, \(k\) is a proportionality constant, and \(T_S\) is the temperature of the surroundings.
Solve this differential equation for a hot cup of coffee that is initially \(80^\circ\)C and is left in a room that is \(20^\circ\)C. In order to find \(k\), you’ll need one more piece of info: suppose the coffee is \(40^\circ\)C after 10 minutes.
Graph the slope field for this differential equation. Could we have predicted the limit of \(T(t)\) as \(t \rightarrow \infty\) without solving?
Another good source of example differential equations comes from electrical circuits. If a circuit has a resistor and an inductor, then the current \(I\) flowing through the circuit obeys
\[L\frac{dI}{dt} + RI = V\]
If \(L = 4\) henrys, \(R = 5\) ohms, and \(V = 6\) volts, solve for \(I(t)\).
Find \(\lim_{t \rightarrow \infty} I(t)\).
Suppose that a population of wild rabbits would be growing exponentially at a 5% rate except that the rabbits are being preyed on by predators at a constant rate. A differential equation for this is: \[\frac{dP}{dt} = 1.05 P - 105.\]
Today we talked about Euler’s method for approximating solutions to differential equations. Here is a Kahn Academy Video that explains the concept. I showed two options for using a computer to do Euler’s method:
Option 1. Use a spreadsheet - You are welcome to do this on the homework if you want.
Option 2. Write a program - This is even easier, if you know just a little Python.
I did several examples using Sage/Python online (see the Examples page). I recommend seeing if you can tweak those examples for the homework this week.