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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.
a4pt 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.
b6pt 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.
a8pt Find the (explicit or implicit) solution for x 5 dy
2
−
= 0, y (1) = π .
dx cos y b10pt Solve xy + 2y − 1
= 0, y (1) = 1.
x2 6 c10pt 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
a6pt Draw the phase diagram and solution curves for
dt
and decide whether they are stable. 8 b10pt 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 c6pt Use improved Euler method with stepsize 1 to estimate y (2) for 10 dy
= x2 − y, y (0) = 1.
dx Problem 4.
a8pt Solve 2y + y + y = 0, y (0) = 0, y (0) = 1 11 b8pt Write the complex number 1 − i in polar form. c10pt Find the general solution of the equation y (6) + 2y (4) + y = 0. 12 ...
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 Fall '07
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