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.
- Find the general solution of y' = 1.
- Find the general solution of y' = 1 + t.
- Find the general solution of y' = 1+y.
- Find the general solution of y" = 1.
- Find the general solution of y" = y.
- Find the eigenvalues of the 2 by 2 matrix
( 1 2 )
( 3 4 )
- On the screen you see a curve in the plane of the form
(x(t),(y(t)) defined for 0 <= t <= 10. Sketch the graphs of
x and y as a function of t.
- Give the formula for approximating a solution to a first
order differential equation of the form y' = F(t,y) by Euler's
method.
- Find the inverse Laplace transform of the function
Y(s) = {s+1}/{s^{2} + 2s +5}.
- State the Existence and Uniqueness Theorem for first order
differential equations.
2. [10 Points] Prove that the Laplace transform of the
function sin(bt)
is Y(s) = b/(s^{2} + b^{2}). 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+x^{2}}
y' = bx(1- {y}/{1+x^{2}})
where b>0 is a parameter.
- Find the equilibrium point for this system and prove that
its location does not depend on b.
- In an essay of no more than two pages, describe the
bifurcation that occurs in this system as the parameter b is
varied. Include a mathematical description of the bifurcation (some
phase portraits) as well a discussion
of what this bifurcation implies in terms of the chemical reaction.
(You need not compute anything but the equilibria for this problem,
i.e., no linearizations are necessary, but see the next question.)
- Five extra points: Compute the linearization and
determine precisely where the bifurcation in part b occurs.
5. [10 Points] Use graphical iteration to describe the fate of a
representative collection of orbits
of F(x) = -x^{3} 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(-x^{2}-y^{2} +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.