Maple Tutor
Part 11: Symbolic
solution of differential equations
In this part we explore
Maple's ability to solve the logistic equation
dy/dt = y (1
- y)
and to check the
solution. Then we will adapt the solution procedure to an initial value
problem with this same differential equation. In the next part, we will
relate these algebraic calculations to the geometry of direction fields.
- Give the differential equation
a name by entering
DE1 := diff(y(t), t) = y(t)*(1 - y(t));
Then ask Maple to solve the equation for y by
entering
dsolve(DE1, y(t));
Note that Maple uses _C1
to represent an arbitrary constant.
- Differentiate your solution
expression with respect to t to get an explicit expression for dy/dt.
Then use your solution expression to find an explicit formula in t
for y(1 - y). Is this formula the same as the one for dy/dt?
You may want to simplify the output before you try to answer this.
- Now we add the initial
condition y(0) = 0.1 to determine a single solution of the differential
equation. To tell Maple to solve the intial value problem, put the
two parts of the problem -- the differential equation and the initial condition
-- in a list:
dsolve ( { DE1, y(0) = 1/10 }, y(t) );
- What do you have to do
to check the answer from the preceding step? Have you done it already?
If not, can you get the checking technique from what you did in Step 2?
- Before you move on to the
next part, define a function h(t) to be the solution of the initial
value problem in Step 3. We will use this function in the next part.