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