MAT 331 Homework Exercises. Week 6 (Oct 28, 99).

#21 (exp. 11/9)

[No Maple] Find all the solutions to the differential equation

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

Among them, single out the one for which x(0) = 3.

#22 (exp. 11/9)

Have Maple solve the following system of differential equations,

$$\left\{\vphantom{ \begin{array}{rcl} y''(x) - z(x) &\!=\!& e^x, \\ z'(x) - y(x) &\!=\!& 0, \end{array} }\right.$$$$\begin{array}{rcl} y''(x) - z(x) &\!=\!& e^x, \\ z'(x) - y(x) &\!=\!& 0, \end{array}$$ $$\left.\vphantom{ \begin{array}{rcl} y''(x) - z(x) &\!=\!& e^x, \\ z'(x) - y(x) &\!=\!& 0, \end{array} }\right.$$

with initial conditions: y(0) = 1, y'(0) = 0, z(0) = k. Let us denote the solutions by y_k(x), z_k(x) (since they depend on the parameter k). For k taking all integer values from -10 to 10, and x $\in$ [- 4, 2], plot on the same graph the functions y_k, using a color, and the functions z_k, using a different color. [This is a case when you don't want to retype the functions that Maple finds. See prob. #06 for hints.]

#23 (exp. 11/9)

Find all the fixed points of the system

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

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.

#24 (exp. 11/9)

Consider the differential equation $\dot{\mathbf{z}}$(t) = F(z(t)), where the vector z(t) = (x(t), y(t)) and the field F(x, y) = (- y, x - y). Plot a few solutions. What happens to them when t$\to$ + $\infty$? Give a ``Maple-proof'' that this is a general fact for every solution. [A ``Maple-proof'' is an argument that is rigorous once we accept Maple results as incontrovertibly true.]

#25 (exp. 11/9)

[No Maple] For the equation $\dot{\mathbf{z}}$ = F(z), z = (x, y), with field

F(x, y) = $$\left(\vphantom{ -x(x^4+y^4)-y \, , \, x-y(x^4+y^4) }\right.$$ - x(x⁴ + y⁴) - y , x - y(x⁴ + y⁴)$$\left.\vphantom{ -x(x^4+y^4)-y \, , \, x-y(x^4+y^4) }\right)$$,

prove that the origin is an attractor in the future, i.e., every solution verifies
[4] $\lim_{t \to +\infty}^{}$z(t) = 0. [Yes! You can ask around how to do this, but then you have to show clearly that you have understood it.]

NOTE: The fact that we use various notations for differential equations is purely intentional.