3f-fall2010-exam_2_sample

# 3f-fall2010-exam_2_sample - Math 3F Exam #2 Sample Laney...

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Math 3F — Exam #2 Sample Laney College, Fall 2010 Fred Bourgoin Disclaimer: This probably is too much to do in 80 minutes, and it might not cover all of the topics you are responsible for. 1. Suppose you know that both y 1 = cos 2 t and y 2 = sin 2 t are solutions to a certain homogeneous second-order linear diﬀerential equation with constant co- eﬃcients. (a) Show that y 1 and y 2 are linearly independent. (b) Is it possible to ﬁnd a third solution that is linearly independent from y 1 and y 2 ? (A ‘yes’ or ‘no’ answer is suﬃcient here.) (c) What is the general solution to the equation? 2. Find the general solution of each homogeneous diﬀerential equation. (a) y ′′ + 2 y + y = 0 (b) 2 y ′′ 3 y + y = 0 (c) y ′′ 2 y + 2 y = 0 3. For each nonhomogeneous equation, give the form of a particular solution Y ( t ). Do not solve for the constants. (Notice that these equations are related to the equations of Problem 2.) (a) y ′′ + 2 y + y = (2 t + 1) e t (b) 2 y ′′ 3 y + y = e t sin 2 t (c) y ′′ 2 y + 2 y = t 2 cos t 2 4. A spring-mass system behaves according to the equation u ′′ + γu + u = 0, where γ is the damping constant. (a) For what value of γ is the system critically damped? (b) For what value(s) of γ is the system overdamped? 5. The charge function Q ( t ) os an electrical circuit satisﬁes the initial-value prob- lem Q ′′ + Q + 4 Q = 3 cos 2 t, Q (0) = 2 , Q (0) = 0 . Find Q ( t ). 1

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6. A mass of 4 kg stretches a spring 0.1 m. If the mass is pushed upward, contract- ing the spring a distance of 0.02 m, and then set in motion with a downward velocity of 1 m/sec, and if there is no damping, ﬁnd the position u ( t ) of the mass at any time t . What is the natural frequency ω 0 of the motion? (For
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## This note was uploaded on 01/14/2011 for the course MATH 3F taught by Professor Williamson during the Fall '09 term at Laney College.

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3f-fall2010-exam_2_sample - Math 3F Exam #2 Sample Laney...

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