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MATH 376: Ordinary Differential Equations
SAMPLE MIDTERM II
Fall 2012
NAME :
NOTE : There are 5 problems on this midterm (total of 6 pages). Use of
calculators to check your work is permitted; however, in order to receive full
credit for any problem, you must show work leading to your answer. You
have 75 minutes to complete this test.
Problem Possible
points
1
20
2
15
3
30
4
15
5
20
Total
100
Score
MATH 376 – SAMPLE MIDTERM II
Problem 1. (20pts) Solve the initial value problem.
y 00 − 6y 0 + 13y = 0,
y(0) = 0, y 0 (0) = 1
Page 2
MATH 376 – SAMPLE MIDTERM II
Page 3
Problem 2. (15pts) Set up the appropriate form of a particular solution yp ,
but do not determine the values of the coefficients.
y 00 − 6y 0 + 13y = xe3x sin 2x
MATH 376 – SAMPLE MIDTERM II
Page 4
Problem 3. (30pts) Find the general solution, then sketch a typical solution.
0 x
4 1
x
0 =
y
6 −1 y
BONUS(+5pts): For which initial conditions (x0 , y0 ) will the solution to
the initial value problem satisfy limt→∞ (xt , yt ) = (0, 0).
MATH 376 – SAMPLE MIDTERM II
Page 5
Problem 4. (15pts) Transform the differential equation into an equivalent
system of first-order differential equations and write your answer in matrix
form.
t3 x(3) − 2t2 x00 + 3tx0 + 5x = ln t
MATH 376 – SAMPLE MIDTERM II
Page 6
Problem 5. (20pts) Apply the Improved Euler’s method with a step size
h = 0.2 to approximate the solution to the following initial value problem on
the interval [0, .4].
y 0 = −2xy,
y(0) = 2