Exam1Fall94 - 65.2400 - 20, 21, 22 TEST 1 Sept. 28, 1994...

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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 _ 62-29(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 - filo/alff/n n,,. 1/1 - u . i/I/n/ul,F/fl,f,! n,. r . .f/f/ff. ¢/v,./-,f,-/‘/-//A,- , , r «n- .Ku/r ff /., ., f f‘,o/a/././n/,fl,f, -/ n], f/fi/f/f/f/nll n/l n/- P f,¢/¢/a/f,¢/,u///-/ / , /4/¢//./f,,./-/u/fl/r/ -/ -/ I, y .In/lfl . quations and plots of solutions of two he corresponding differential equation and explain wh e l m t n e H Wm d r u b .u..pu..--vu.p..lh,n- e ...>J--v.-..,....in- ....,svov.....4.h.\-~ .. / . i wwmym 12,:/....1.:Ir._§ bmm H 4 ...j...;,.,././,..,.,,W\‘\..\\.i.\. 1%..% C lU/u / ...4,_t./...,.r.,. y.\.fi.\.\- .1 ) , ,.,,., .\ T \\ n..E _ y ...,.,,,.., 1/.» ,.‘..w.\\\. mte 1 _ . imm M1 .1: L; . G h 1 1 y“ {is Ammo _ _ _ y ,, C; tar 4 _ : i, 8W = __ {K 4 \._: a, .//.,,, Pmy y y _ ( 51:7 7,, 5Fm + _ __ __ 1.7? I, 1 / I I I _ y y y y 3.: /,/ . e . . . 3Mb A B C D / tnu “a £10 _ oy N i .\ / \\. Equation: Explanation: Equation: 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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Exam1Fall94 - 65.2400 - 20, 21, 22 TEST 1 Sept. 28, 1994...

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