Math 331, Fall 2004: Problem Set 5. Due Oct18

1. 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.]

2. Consider the first order system
   
   $$\left\{ \begin{array}{rcl} x'&\!=\!&-2x+3y\\ y'&\!=\!&-3x \end{array} \right.$$
   
   
   • a Find the fixed points.
   • b Determine the 'shape" of the solutions around the fixed point (that is, study whether it is a saddle, a source, a spiral source, a sink, a spiral sink or a center)
   • c Determine whether
     
     $$\left\{ \begin{array}{rcl} x(t)&\!=\!&e^{-t}\cos(2\sqrt{2}... ...}t))\\ y(t)&\!=\!&3e^{-t}\cos(2\sqrt{2}t) \end{array} \right.$$
     
     
     a solution of the system. Justify your answer.

3. (no Maple)Consider a linear system $X'=AX$, where $A$ is a real 2x2 matrix. Show that if $V(t)=(v_1(t),v_2(t))$ and $U(t)=(u_1(t),u_2(t))$ are two solutions and $r$ and $s$ are two real numbers, then $W(t)=r V(t) + s U(t)$ is also a solution.

4. 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.