### Sample Final II - Department of Mathematics

DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KANSAS
MATH 220 - FALL 2009 - FINAL EXAM
On this exam, you may use a calculator and formula notes.
It is not sufficient to just write down the answers.
You must explain how you arrived at your answers and how you know they are correct.
1
(35)
2
(35)
3
(35)
4
(35)
5
(35)
6
(35)
7
(35)
8
(35)
9
(35)
10
(35)
Total (350)
2
• 1. (35 points) Solve the initial-value problem and sketch the graph of the
solution
2t
y0 + 2
y = 2t(t2 + 1), y(1) = 2.
t +1
• 2. (35 points) Find the solution in explicit form and determine the interval
of existence of the solution for the initial-value problem
t + ye−t y 0 = 0,
y(0) = 1.
3
• 3. (35 points) Consider a tank used in certain hydrodynamic experiments.
After one experiment the tank contains 200 liters of dye solution with a concentration of 1g/liter. To prepare for the next experiment, the tank is to be
rinsed with fresh water flowing in at a rate of 2 liters/min, the well-stirred
solution flowing out at the same rate. Find the time that will ellapse before
the concentration of dye in the tank reaches 1 % of its original value.
4
• 4. (35 points) Find the solution of the initial-value problem
y 00 + 2y 0 + 5y = 0
y(0) = 1
0
y (0) = 0
• 5. (35 points) Find the general solutions of
y 00 + 4y = t2 + 3et
5
• 6. (35 points) Solve the exact equation
(9x2 + y − 1)dx − (4y − x)dy = 0,
y(1) = 0
and determine where the solution is valid.
• 7. (35 points) Find the general solution of the linear system
0
x = x+y
y 0 = 4x + y.
6
• 8. (35 points) Find the solution of the initial-value problem
0
x1 = x1 − 4x2
x02 = 4x1 − 7x2 .
where
3
x(0) =
.
2
Draw the graph of the solution and describe its behavior for increasing t.
7
• 9. (35 points) Use Laplace transform to solve the initial-value problem
y 00 + 2y 0 + 5y = sin 2t
y(0) = 2
0
y (0) = −1
• 10. (35 points) Solve the system of equations
0
x1 = 2x1 − 5x2
x02 = x1 − 2x2