MA 231: Honors Differential Equations

Final Exam

Do all problems on the blank sheets in order. Please write the problem number in the upper right hand corner of each sheet. Be sure to show all work.

1. [20 Points] Quickies. Answers only---no partial credit.

2. [10 Points] Prove that the Laplace transform of the function sin(bt) is Y(s) = b/(s2 + b2). Show all steps.

3. [20 Points] Find the general solution of the linear system Y' = AY where A is the matrix
( 0 2 )
(-8 0 )
Then sketch the phase portrait for this system.

4. [10 Points] Consider the system governing the chemical reaction involving chlorine dioxide and iodide given by

x' = -x +10 -{4xy}/{1+x2}
y' = bx(1- {y}/{1+x2})
where b>0 is a parameter.

5. [10 Points] Use graphical iteration to describe the fate of a representative collection of orbits of F(x) = -x3 for seeds -1 <= x <= 1. Find all fixed points and 2-cycles for this function.

6. [10 Points] Solve the differential equation y" + y = 2 if t < 4, 0 if t >= 4. with y(0)=0 and y'(0) = 0 using Laplace transforms. Show all work.

7. [20 Points] Consider the system of equations defined for x,y >= 0.

x' = x(-x-y+40)
y' = y(-x2-y2 +2500).
First find all equilibirum points. Determine the type of these equilibrium points using linearization. Sketch the nullclines and indicate the direction of the vector field on and in between these curves. Finally, sketch the solution curves of this system. Remember that x, y >= 0.