Math 331, Fall 2002: Problems 13-16

NOTE: Each exercise is worth 10 points and can be turned in at any time before its ``expiration date''. At the end of the semester, I will expect you to have turned in at least 2/5 of the exercises assigned. If you do more, I will pick your best grades. If you do less, the missing grades will be counted as zeros. Altogether, these will count the same as one project.

13.

(expires 10/20)     Find all the solutions to the differential equation

$$\frac{dx}{dt}(t) = -2 x(t), \quad t\in{\mathbf{R}}.$$

Among them, single out the one for which $x(0)=3$. [Hint: read the help page for dsolve, or just do it in your head. It is that easy.]

14.

(expires 10/20)     Have Maple find analytic solutions to the following system of differential equations,

$$\left\{ \begin{array}{rcl} y''(t) - z(t) &\!=\!& e^t, \\ z'(t) - y(t) &\!=\!& 0, \end{array} \right.$$

with initial conditions: $y(0)=1,\ y'(0)=0,\ z(0)=k$. Let us denote the solutions by $y_k(t),\ z_k(t)$ (since they depend on the parameter $k$).

For $k$ taking all integer values from -10 to 10, and $t \in [-4,2]$, plot the functions $y_k$ in blue, and the functions $z_k$ in red, all on the same graph. (Yes, you will then have 42 functions plotted on the same graph.) [This is certainly a case when you don't want to retype the functions that Maple finds. You will almost certainly need to read the help page for dsolve. I also found subs, unapply, and seq useful.]

15.

(expires 10/20)     For the functions $y_k(t)$ and $z_k(t)$ found in problem #14, plot the parametric curves $\varphi_k(t) = [y_k(t), z_k(t)]$ for integer values of $k$ between $-5$ and $5$ and $-6 < t <4$ on the same graph. Use the view option of plot to only show what lies in the region $-10<y<10, -10<z<10$, and use a sequence of colors so that each solution is a different color. [ HINT: you might find something like seq(COLOR(HUE,i/11),i=0..10) useful for the latter.]

16.

(expires 10/20)     Find all the fixed points of the system

$$\left\{ \begin{array}{rcl} \dot{x} &\!=\!& x^2 + y, \\ \dot{y} &\!=\!& x (y^2 - 1), \end{array} \right.$$

a fixed point being a solution for which both $x(t)$ and $y(t)$ stay constant. For each of these points, describe the behavior of the solutions that have initial conditions nearby. You can use Maple to figure out what happens for nearby points, or you can use more mathematical methods.

NOTE: The fact that there are various notations for differential equations is purely intentional.