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Helpful Hints for Finding Differential Equations:
The following are a couple of examples. Keep these hints in mind as you work through the problems.
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#1- Loss of Knowledge (Taken from the worksheet in class 11-12-99)
"Our book suggests facetiously that students forget material after they have learned it. One model for this assumes that the rate at which a student forgets material is proportional to the difference between the material he or she currently remembers and some positive constant, a." LaRose.
1. What should the dependent variable be? What units would you attach to it?
-The dependent variable in this case would be knowledge and we can assign any letter to it, say N. We can attach any units that seem appropriate. For our use we will assign the units of grade points. We will measure our knowledge in grade points earned.
2. What rates affect this? Write the differential equation.
-Two different rates effect this. First of all, the rate of change in knowledge with respect to time. This can be written as dN/dt. Second, there is the rate at which knowledge is lost. This can be written as k(N-a), where k is a constant. Our differential equation would simply be:
3. Solve the differential equation. What is the significance of the parameter a in your solution?
-To solve the differential equation, we first get the same variables on the same side (we need all N's on the left and all t's on the right).
dN/(N-a) = dt k
We then take the antiderivative of each side and get
ln(N-a) = tk + c
Next, we want to solve for N, first by getting rid of the natural log, then by adding a to both sides.
(N-a) = e^(tk + c)
N = a + qe^(tk), where q and k are constants.
(We got q by splitting up e^(tk + c) into e^(tk)e^(c). Since c is a constant, e^(c) is a constant and can be renamed q. Remember that we're being sneaky about absolute values and positives and negatives here.).
4. What is the significance of the parameter a in your solution?
-We need a because there is only a certain amount of information you can lose. Therefore, a is that amount of knowledge that it's impossible to lose.
#2- Exporting Woozles
"Suppose we are raising woozles for export. The number of woozles increases due to the reproduction at a rate proportional to the number present. It also decreases at a rate proportional to the number squared because of overcrowding, and also decreases at a rate of E woozles/day from exports."
1. What is the dependent variable and what are units used?
-The dependent variable in this case is the number of woozles, say N. Our units are actually the number of woozles.
2. What are the rates?
- Our input rate is the number of woozles as a result of reproduction.
Input Rate = k N (where k is the proportional constant) woozles/day
-Our output rates (decrease rates) are the number of woozles lost because of overcrowding and export.
Overcrowding = m N^2 woozles/day
Exporting = E woozles/day
3. What then, is the differential equation?
-We can find the differential equation very easily by adding the input and subtracting the output.
dN/dt = k N - m N^2 - E
4. Hmm. Can we solve this?
-Let's plug in a few numbers first. Let's have m = E = 1 and k = 2. Then...
dN/dt = -N^2 + 2N - 1 = -(N^2 - 2N + 1) = -(N - 1)^2
-Now we must solve the equation using the same steps as the above example.
dN/(N - 1)^2 = -dt (antidifferentiate this)
-1/(N - 1) = -t + c
N - 1 = 1/(t + c) {Note: This c and the c from the previous step are two different c's. c= -c}
N = 1 + 1/(t + c) where c is a constant.
We would appreciate your anonymous response so that we may be able to help you more in the future.
Very helpful, I can definitely use this information, and it helped to clear up any confusion.
Somewhat helpful, although I'm still a little confused.
Not helpful at all.