# midterm1 - t=10 4 Problem 3 a Let dx dt =-5 x 2 x 3 Find...

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Name: Section/Time of lecture: Professor/GSI: MATH 216 MIDTERM I Each part of a problem counts equally. To get full score you need to carefully explain what you did. No calculators allowed. 1

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Problem Points Score 1 30 2 30 3 30 4 60 TOTAL 150 2
Problem 1. a) Verify that the function y ( x ) = 1 x + ln x satisﬁes x 2 y 00 + 2 xy 0 - xy = - x ln x b) Solve dy dx = sin x (cos x ) 2 + 1 x 2 , y (2 π ) = 1 . c) Sketch the slope ﬁeld and solution curves for dy dx = 1 - x 2 where deﬁned. Include enough detail to be able to give a rough sketch of the solution curves with y (0) = 0 and y (0) = 1 . 3

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Problem 2. a) Solve dy dx = x (1 - 2 y ) e 2 x , y (0) = 2 b) Solve ( x 2 + 1) y 0 = - 3 xy + x, y (0) = 0 . c) The time rate of change of a population P is proportional to the cube root of P. At time t=0, the population is 1000 and the rate of increase is 100 at that time. What is the population at time

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Unformatted text preview: t=10? 4 Problem 3. a) Let dx dt =-5 x + 2 x 3 . Find the equilibrium solutions. Which ones are stable? b) Suppose that a motor boat is moving at 100 m/s when the motor quits. After 20 seconds, the boat has slowed to 10 m/s. Assume the resistance is proportional to its speed. Hence we have the diﬀerential equation for the speed v : dv dt =-kv. How far will the boat coast? c) Use Euler with stepsize . 5 to estimate y (2) for dy dx = x 2 + y 3 , y (1) = 0 . 5 Problem 4. a) Solve the equation y 00-3 y-10 = 0 with initial conditions y (0) = 3 , y (0) =-1 . b) Solve the equation 5 z 2-2 z + 10 = 0 . c) Solve 9 y 00 + 6 y + 25 y = 0 , y (0) = 2 , y (0) = 1 6...
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## This note was uploaded on 01/18/2009 for the course MATH 216 taught by Professor Stenstones? during the Spring '07 term at University of Michigan.

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midterm1 - t=10 4 Problem 3 a Let dx dt =-5 x 2 x 3 Find...

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