13 Ordinary Differential Equations V

1. Solve the system
   
   $${\vec x} ' = A \vec x \qquad A = \left( \begin{array}{cc} 3 & -5 \ 8 & -3 \end{array} \right)$$
   
   
   Neatly and accurately sketch a comprehensive set of solution curves in the $x_1,x_2$ plane. Include the eigenvectors, $\vec f$, $\vec u$, and $\vec v$ in the graph if applicable. Get the slopes right.

2. Solve the system
   
   $$\dot {\vec x} = A \vec x \qquad A = \left( \begin{array}{cc} -6 & -7 \ 7 & -20 \end{array} \right)$$
   
   
   Neatly and accurately sketch a comprehensive set of solution curves in the $x_1,x_2$ plane. Include the eigenvectors, $\vec f$, $\vec u$, and $\vec v$ in the graph if applicable. Get the slopes right.

3. (30 pt).

   Please answer in order asked (even if you do the numerics first to guard against errors).

   Consider the autonomous system
   

   $$x' = x + 3y - x^2 \sin y \qquad y' = 2x + y - x y^2$$
   
   

   First analyze this system analytically:

   1. Find the critical points. One critical point is easy. Four more critical points can be found numerically. To help you a bit, their $y$-values are $\pm1.1107$ and $\pm1.6074$.
   2. Find the matrix of derivatives of vector $\vec F$ at each of the five critical point. (Actually, you can use symmetry around the origin and only find three.)
   3. Use it to analyze each critical point. List type of point and its stability.
   4. Also find the relevant eigenvectors or $\vec u$ and $\vec v$ if complex, and a $\vec f$ if defective.
   5. Sketch the solution lines in the immediate vicinity of each critical point in a single $x,y$ plane. Take $-5\le x\le5$ and $-5\le y\le5$. Try to get it as accurate as possible. To do so, use the directions of the eigenvectors, $\vec f$, or $\vec u$ and $\vec v$.
   6. State whether critical point analysis must give the right solution near the point. Could the real lines be qualitatively different from what you drew?

   Next draw the complete solution plane as a dense and complete set of solution lines using a numerical solution method.

   Suitable programs to do this can be found on the web. Some I saw previously:

   http://www.math.uu.nl/people/beukers/phase/newphase.html

   http://www.scottsarra.org/applets/dirField1/dirField1.html

   http://www.math.rutgers.edu/courses/ODE/sherod/phase-local.html

   See here for Matlab software. (You will need to convert to an ODE by taking the ratio of the equations, and then the software might crash when it divides by zero if it hits a critical point.)

   You will need to use a screen grabber to make a copy that you can print. Typically you press Alt+PrintScreen or Shift-PrintScreen to get a printable copy of the active Window.

   Comment on how well your predictions came out.