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Unformatted text preview: 65.2400  20, 21, 22 TEST 1 Sept. 28, 1994 Name Section 20 21 22 Please answer all questions, showing your work in detail. You may refer to one sheet (8.5 by 11 inches) of notes. No other notes, books, references,
calculators, etc. are permitted. NAME _______________________________ __ SECTION ____________________ __ 1. (a) (15 points) Solve the initial value problem d
~y»2y=e“z—1, y(0)=A.
dr Find the value of A that separates solutions that grow large and positively as 1: —> 00
from those that grow large and negatively. 1. (b) (10 points) Solve the initial value problem d_y _ 6229(3 — 6x2) _.—————— 1:,
d3: 2 ,y()0 Express the solution in explicit form, that is, y = NAME _______________________________  SECTION ____________________ __ 2. (a) (15 points) Find the solution of the initial value problem dgy dy 5
~ — — = r = 1. ’ = 2.
dxg + dx + 2y 0, y(0) 21(0) How does the solution behave for large values of r? 2. (b) (15 points) A hospital patient is receiving medication intravenously at the rate of
k milligrams per hour. The medication is absorbed from the bloodstream into body
tissues at a rate of twenty percent per hour. Let M (t) be the amount of medication
in the patient’s bloodstream at time t. i. Write down a differential equation satisﬁed by M
ii. Determine the equilibrium level of medication in the patient’s bloodstream. iii. Determine the medication rate k that is needed to maintain a constant level of 150
milligrams of medication in the bloodstream. SECTION NAME _____________ o I f f I, t I I u 
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. quations and plots of solutions of two he corresponding differential equation and explain wh e
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Explanation: LAB TEST 1 65.2400 — 20, 21, 22 September 30, 1994 NAME _______________________________ __ SECTION 1. Consider the initial value problem (a) Use dsolve to solve this problem Write the solution in the space below. (b) Plot the solution corresponding to A = —l for 0 g t g 20. Sketch its graph on the
axes below. ti (c) For the solution with A = 1 determine the ﬁrst positive value of t for which y is zero.
t =
(d) For the solution with A = —1 determine the coordinates of the minimum point on the
graph. t=_____ y: '3, Consider the llllft"l“‘lirltil equation _. {/2 ‘ \I 4
——— — e
(if '2 (a) Draw a direction ﬁeld for this equation on the rectangle l) g t g 8. —4 g y g 4.
Describe in a sentence or two how the solutions appear to behave as t increases. (b) For large 1‘ solutions behave in two different ways depending on whether the initial value y(0) is greater than or less than a certain critical value yo. Estimate this critical
initial value. “ 90 ~ .__..._.__. (c) Plot the solutions that satisfy the initial conditions y(0)
tivelyi Sketch their graphs on the axes below. ~1.5 and y(0) = 4, respec (d) From the graph, estimate the coordinates of the minimum point for the solution start
ing at y(0) = 4. ‘ t:_________ y: ...
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 Spring '04
 Yoon
 Differential Equations, Equations

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