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MATH 251
Summer 2002 Exam 1 June 27, 2002
NAME :
ID :
INSTRUCTOR :
There are 10 questions on 9 pages. Please read each problem carefully before starting to solve it. For each multiple choice problem 4 answers are given, only one of which is correct. Mark only one choice. For partial credit questions, all work must be shown - credit will not be given for an answer unsupported by work.
NO CALCULATORS ARE ALLOWED.
PLEASE DO NOT WRITE IN THE BOX BELOW.
Total:
- (6 points) Which of the following is a linear second order differential equation? (a) y′^ + ty = 1 (b) y′′^ = t^2 y + et (c) (y′)^2 = (y + 2)(y − 3) (d) y′′^ + 3y′^ + 2y = sin y
- (6 points) The Existence and Uniqueness Theorem guarantees that the solution to
(t^2 − 4)y′^ + 2ty = t^12 y(1) = 4
is valid on; (a) (− 2 , 2) (b) (0, 2) (c) (2, ∞) (d) (−∞, ∞)
- (12 points) Solve explicitly the initial value problem;
y′^ =^3 x
(^2) + 4x + 2 2 y − 6 y(0) = 4
- (14 points) A 400-liter tank is initially filled with 100 liters of dye solution with a dye concentration of 5 g/l. Pure water flows into the tank at a rate of 3 liters per minute. The well-stirred solution is drained at a rate of 2 liters per minute. Find the concentration of dye in the tank at the time that the tank is completely filled.
- (12 points) Solve the initial value problem; ( 2 x + 4xy + y x
) dx + (2x^2 + ln x + 3y^2 )dy = 0 y(1) = − 1
- (14 points) (a) Solve the initial value problem: y′′^ − y′^ − 6 y = 0, y(0) = α, y′(0) = 4. (b) If limt→∞ y(t) = 0, what is the value of α?
