FINAL EXAM

Differential Equations

Fall 2004, Hunsicker

There are 200 points possible on this exam. You have three hours. Complete as much of it as you are able to. Points are indicated for each problem, so allot your time accordingly. SHOW ALL WORK! No books, notes, or conversation is allowed. You may use a calculator, but please indicate where and how you have used it, except for basic numerical calculations. You may not use any calculus functions on your calculator.

Come see me in my office if you have questions or start to panic.

Definitions

1) (10 points) Define exact o.d.e.

2) (15 points) Define Picard iterates and briefly explain what they are and why

we care about them.

3) (10 points) Define quasi-periodic, critically damped, and overdamped spring

systems.

4) (15 points) Define vector space.

5) (10 points) Define basis and dimension.

6) (10 points) Define the determinant of an nxn matrix.

Theorems

7) (10 points) State the existence and uniqueness theorem for linear second

order IVPs.

8) (15 points) Prove Abel's theorem (version for second order equations).

9) (10 points) Prove that in any vector space V and for any element v in V,

0v = 0.

10) (15 points) Prove that the eigenvalues of a matrix M are the roots of the

equation det(M-lI)=0.

11) (10 points) State the subspace theorem.

12) (10 points) State the theorem from the book that is equivalent to saying that

the solution space of a system of n linear equations is n dimensional.

Problems

13) a) (5 points) Draw the slope field diagram for the equation y' = x / (1+y).

b) (10 points) Solve the equation above given the initial condition y(0)=2.

14) (10 points) A population of bacteria growing in a petri dish with ample food

and space has an initial mass of .05 grams. After 3 days, it has a mass of

1.25 grams. Assuming that it grows at a rate proportional to its mass at

any given time and that food will not become scarce within 5 days, how

much will it weigh at the end of that time?

15) (10 points) When a 250 gram weight is suspended from a spring, the spring

stretches by 10 cm. Assume the system is suspended in a vat of peanut

oil which has a viscous damping constant of 500 dyne-sec/cm. If the

weight is pulled down an additional 5 cm and released, find the function

that describes its motion. Recall that acceleration due to gravity is 9.8

meters/s².

16) (10 points) Three tanks of sugar solution are connected by tubes and

pumps. The first tank contains 100 gallons of solution, the second tank

contains 75 gallons of solution and the third tank contains 50 gallons of

solution.

Solution is pumped from the first tank to the second tank at a rate of 10

gal/hour. Solution is pumped from the first tank to the third tank at a rate of

2 gal/hour. A solution with 2 lb of sugar/gal is added to the first tank from

an external source at a rate of 6 gal/hour.

Solution is pumped from the second tank to the third tank at a rate of 6

gal/hour and drained to the outside at a rate of 4 gal/hour.

Solution is pumped from the third tank back to the first tank at a rate of 5

gal/hr and is drained to the outside at a rate of 1 gal/hr.

Draw a diagram, then set up the system of 3 linear differential equations

which describes this situation. YOU NEED NOT SOLVE!

17) (10 points) Find the determinant of the matrix

0 -1 0 -2

1 2 -1 0

-1 1 0 1

0 1 1 0

18) a) (10 points) Solve the system:

x' = x + y

y' = 4x - 2y

b) (5 points) Draw the trajectories of this system in the phase plane.