Download Math 232 – Winter 2003

MATH 232: Final Exam, 18 March 2003
Instructions: Do 9 of the following 10 problems. Please show all appropriate work. Good luck
and have a great spring break!
1. Solve the differential equation
dy
 ty 2  2 y 2 subject to y(0)=1.
dt
2. (a) Use Euler's method with step size of .25 to estimate to solution to
dy
 ( y  5) 4
dt
at t = 1. Given that y(.5) = 1.
b) Bob claimed that every solution to will increase to infinity no matter what the initial
condition is because the derivative is always nonnegative. Without solving the equation,
determine whether Bob is correct, and justify your argument.
dP
P2
3. Consider the population model
that models a species of fish in a pond where t
 2P 
dt
50
is measured in years.
a) Describe the long term behavior of the population (assuming that the initial population is
positive).
b) Modify the equation to represent a model where k fish are harvested each year.
c) What is the largest number of fish that can be harvested each year, if the fish are to have a
chance to survive?
4. For this problem, consider the differential equation.
a) Use the substitution u = y – t to rewrite the equation
dy
2
 (y  t)  3(y  t)  3
dt
as an autonomous differential equation, and then sketch the phase line for
the autonomous differential equation.
b) Sketch the equilibrium solutions of the original equation on a graph and then
c) use (a) and (b) to provide a rough sketch for solutions to the original differential
equation that satisfy the initial conditions: (i) y(0) = 0
(ii) y(3)=0
(iii) y(0) = 4.
5. Consider a system of linear differential equations
dY  3 1 

Y
dt  3  1
a) Given that A has eigenvalues and eigenvectors as below, match the eigenvalues with the
eigenvectors and write the general solution to the system.
1  1
Eigenvectors:  ,  
3  1 
Eigenvalues: -4, 0.
b) Sketch the phase portrait for the system.
c) What is the long term behavior of a solution that starts at the point (0,4), be as explicit as you
can be.
 4 2
6. (a) Let A  
, find the eigenvalues and eigenvectors of A.
1 3
(b) Find the solution to
dY
 AY that passes through the point (4,-1) where A is as in (a).
dt
d 2y
dy
 8y  2e3t .
2 6
dt
dt
b) What form of the particular solution is needed to solve
7. a) Find the general solution of
d2y
 4 y  t sin( 2t )
dt 2
using the method of undetermined coefficients. You do not need to find the particular solution.
8. Consider the system of differential equations for nonnegative values of x and y
dx
 x( x  3 y  150)
dt
and
dy
 y (2 x  y  100).
dt
a) Is this a linear or nonlinear system of differential equations, explain.
b) If this is considered as a population model for two different populations x and y, would you
describe this as a preditor/prey model, or a competing species model? Explain your answer.
c) Find the equilibrium points for the system.
9. Consider system of differential equations as in question 8.
a) Find and sketch the nullclines
b) Using (a) and an analysis of the signs of the derivatives, sketch a rough direction field for the
system of differential equations in the first quadrant.
10. Consider the ideal pendulum system with bob mass m and arm length l given by
d
dv
g
 v and
  sin  .
dt
dt
l
a) Find the linearization of the ideal pendulum system at the equilibrium point (0,0).
b) Compute the eigenvalues for the linearized system, and describe the type of equilibrium point
at (0,0) of the linearized system.
c) What is the period of solutions for the linearized system (the answer will depend on l and g).
d) Using g = 9.8m/s2, what value of l should be chosen so that the period of solutions to the
linearized system is 1? Would this period be more likely to represent the nonlinear solution with
small swings or large swings? Why?