Math 331, Fall 2002: Problems 17-20

17.
(expires 10/28)     Consider the differential equation , where the vector and the field . Plot a few solutions. What happens to them when ? 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.]

18.
(expires 10/28)    (No Maple.) For the equation , , with the vector field

prove that the origin is an attractor in the future, i.e., every solution verifies

[You can ask around how to do this, but then you have to show clearly that you have understood it.]

19.
(expires 10/28)     For the system of differential equations of prob. #23,

find the eigenvalues and eigenvectors of the Jacobian at the fixed points. [This is a give-away if you have done #16.]

20.
(expires 10/28)     Consider the equations of the glider with no drag term (). Use dsolve, type=numeric to solve them numerically with initial conditions , . Then solve exactly the linearized system around the fixed point , with the same initial conditions. Graph the two functions for , and give a good estimate of their maximum difference. What happens if we take a larger -range?

MAT 331 2002-10-21