Homework 9
In this homework, we're given the values for several constants (parameters) that we will be using
over and over again. Let's start by assigning our values to these parameters here in Maple.
(1)
(2)
(3)
(4)
If you wanted to write down the differential equation for the Logistic model by hand, you would
write "y'=k*y*(M-y)." In Maple, you can type exactly the same thing.
(5)
Notice that Maple makes (population) a function of by default. The problems say that
(population) is a function of (time). This just means that when we see in our problem, we use
here in Maple.
Equilibrium solutions to a differential equation are values of the function where the derivative is
zero. If we wanted to solve for equilibrium solutions of the Logistic differential equation by
hand, one thing we could do is ignore the left-hand side of the differential equation and simply
set the right-hand side equal to 0 and solve for . This turns out to be the easiest way to do the
same thing in Maple.
To grab only the right-hand side of an equation in Maple, use the rhs command. The syntax is
. For example
(6)
We want to set this expression equal to 0 and solve that equation. Remember, use the solve
command to solve an equation!
(7)
What happened? What is this mess? The command
will work for most standard
equations. Maple will search the equation for a variable and solve for that variable. Our
equation involves the variable
, and Maple is trying to solve for , not
like we want.
To tell Maple to solve an equation for a specific variable, simply add that variable to the solve
command:
.
(8)
Now we have our two equilibrium solutions:
solutions are
and
.
and
. Notice that the equilibrium
In order to actually get the Logistic model, we have to solve the differential equation. To solve a
differential equation in Maple, use the
command. The syntax is:
Remember that initial conditions are values when time (or, in our case, ) equals 0.
Let's solve the differential equation and assign the Logistic model to a variable.
(9)
Now that we have our model, we can calculate the population at a given time. To substitute a
value into an expression, use the subs command.
(10)
Maple has given us an exact answer. We would like a decimal approximation so that we can
enter our answer into WebAssign. Use the
command.
(11)
Next, let's enter the differential equation for the Logistic Model with Harvesting
(12)
We can find the equilibrium solutions of this differential equation the same way we found the
equilibrium solutions for the Logistic differential equation.
If we want to sketch solutions to this differential equation, we can use the
command in
the
package. Remember that the colon at the end of the line tells Maple not to output
anything.
250000
200000
150000
y(x)
100000
50000
0
2
4
6
x
8
10
12
WebAssign may ask for two plots, but as long as you show two curves on one plot like this
example, your work will be correct.