# exam1W09 - Name Section/Time of lecture Professor/GSI MATH...

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Unformatted text preview: Name: Section/Time of lecture: Professor/GSI: MATH 216 WINTER 2009 MIDTERM I To get full score you need to carefully explain what you did. No calculators allowed. 1 Problem Points Score 1 10 2 28 3 22 4 26 2 TOTAL 88 2 Problem 1. a-4pt Verify that the function y (x) = x2 sin x satisﬁes y ￿￿ − 3 y￿ + y = 3x cos x. x dy = x + y 2 . Include enough detail to be able dx to give a rough sketch of the solution curves with y (1) = 0 and y (0) = 0.5. b-6pt Sketch the slope ﬁeld and solution curves for y 1.5 1 0.5 −2 −1.5 −1 −0.5 0.5 −0.5 −1 −1.5 4 1 1.5 2 x Problem 2. a-8pt Find the (explicit or implicit) solution for x 5 dy 2 − = 0, y (1) = π . dx cos y b-10pt Solve xy ￿ + 2y − 1 = 0, y (1) = 1. x2 6 c-10pt Let x(t) represent the number of grams of salt dissolved in a water solution. Assume that x(t) satisﬁes x￿ = x − x2 and x = 0.1 when t = 0 minute. How many minutes will it take for an additional 0.1 grams of salt to dissolve? 7 Problem 3. dx = x − 4x3 . Find the equilibrium solutions a-6pt Draw the phase diagram and solution curves for dt and decide whether they are stable. 8 b-10pt Let v (t) denote the speed of a motorboat at time t. The motorboat starts from rest with a constant acceleration of 10f t/sec2 and the water resistance causes a deceleration of v 2 f t/sec2 . Find the limiting velocity of the boat as t → ∞ (that is, the maximum possible speed). 9 c-6pt Use improved Euler method with stepsize 1 to estimate y (2) for 10 dy = x2 − y, y (0) = 1. dx Problem 4. a-8pt Solve 2y ￿￿ + y ￿ + y = 0, y (0) = 0, y ￿ (0) = 1 11 b-8pt Write the complex number 1 − i in polar form. c-10pt Find the general solution of the equation y (6) + 2y (4) + y ￿￿ = 0. 12 ...
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## This note was uploaded on 01/30/2012 for the course MATH 216 taught by Professor Stenstones? during the Fall '07 term at University of Michigan.

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