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MATH Name: 251 Midterm Exam I Feb. 26, 2004 Student Number: Instructor: Section:
This exam has 10 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown. Credit will not be given for an answer not supported by work. THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION. At the end of the examination, the booklet will be collected.
Do not write in this box.
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Total:
MATH 251 Spring 2004 Exam I 1. (5 points) Consider the initial value problem: 1 dy = , dt y - 3t2 y(t0 ) = y0
Give conditions on the values of t0 and y0 so that you are guaranteed to find a unique solution to the problem. Do not attempt to solve the equation for y(t).
Page 2 of 10
MATH 251 Spring 2004 Exam I 2. (12 points) For the autonomous equation y = y 3 + 2y 2 - 3y a)Find all equilibrium solutions.
b) Determine the stability of each equilibrium solution you found in part (a). Justify your conclusions.
c) Let y(t) be the solution whose initial condition is y(0) = 4. What is the behavior of y(t) as t + ?
d) Let y(t) be the solution whose initial condition is y(0) = -1. What is the behavior of y(t) as t + ?
Page 3 of 10
MATH 251 Spring 2004 Exam I 3. (10 points) Given that y 2 ex y + 4x3 + (2x y exy + 2) a) Verify that the equation is exact.
2 2
dy = 0, y(0) = 2 dx
b) Solve the initial value problem. Leave your answer in implicit form.
Page 4 of 10
MATH 251 Spring 2004 Exam I 4. (15 points) For the equation y - 4y - 5y = 0 a) Find a fundamental pair of solutions.
b) Based on a), find the general solution.
c) Find the solution satisfying conditions initial y(0) = 2, y (0) = 3.
Page 5 of 10
MATH 251 Spring 2004 Exam I 5. (a) (6 points) Solve the initial value problem: y = cos t + 1 ey y(0) = 3.
(b) (6 points) Solve the initial value problem t2 y + ty = 2 y(1) = 2.
Page 6 of 10
MATH 251 Spring 2004 Exam I 6. Find general solutions to the following: (a) (6 points) y + 2y + 3y = 0.
(b) (6 points) y + 6y + 9y = 0.
Page 7 of 10
MATH 251 Spring 2004 Exam I 7. (6 points) Indicate whether each equation below is linear or non-linear.
dy (a) sin(x) + ex dx = y 4 .
(b) y + p(t)y + et = 0.
(c) y = y 2 cos(t).
(d) y = cos(y).
(e) y + t4 y + t3 y = 0.
8. (6 points) Indicate whether the two functions are linearly independent on the interval (-, +):
(a) e2t , e2t + 2
(b) sin(2t), sin(2t + 2)
(c) t + 1, 2t + 2
Page 8 of 10
MATH 251 Spring 2004 Exam I 9. (7 points) Given the linear ordinary differential equation ty - 4y + 4e y = 0 and two fundat mental solutions y1 (t), y2 (t) such that y1 (1) = 1, y1 (1) = 0, y2 (1) = 2 and y2 (1) = 3, compute their Wronskian W (y1 (t), y2 (t)) as a function of time, using Abel's Theorem . Use the initial condition to determine the constant of the Wronskian.
t
Page 9 of 10
MATH 251 Spring 2004 Exam I 10. (15 points) A tank initially contains 120 liters of pure water. A salt solution with a concentration of grams/liter of salt enters the tank at a rate of 2 liters/min and the well-stirred mixture leaves the tank at the same rate. Find (in terms of ) an expression for the amount of salt in the tank at any time t and the limiting amount of salt in the tank as t approaches infinity.
Page 10 of 10
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