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Exam1Spring94

# Exam1Spring94 - LAB TEST 1 65.2400 — 07 08 09 February...

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Unformatted text preview: LAB TEST 1 65.2400 — 07, 08, 09 February 1-1. 1994 NAME _._.._...__.____________.._.- SECTION -_._.._____ 1. Consider the initial value problem cit/#1 . . . — t —- =3+2cosl2tl. (0\=O dt 4” y‘ (a) Use dsolve to ﬁnd the solution. Write it in the space below. (b) Plot the solution for ll g f S '30. Skett-h the graph on the axes below. Indicate the scale on the y—axis. U? (0) Determine the value of the solution at t = in U = _______ --_____...__ (d) Determine the value r6 f for which the >olution first inteisects the line y = 12. r: 2. Consider the differential equation dy —- = t" — 2t dt J (a) Is this equation linear? Yes ._ No _ Is this equation separable? Yes _ No _ (b) Draw a direction ﬁeld for this equation on the rectangle 0 _<_ t S 6. —3 S y g 3. Describe in a sentence or two how the solutions appear to behave as t increases. (c) For large t solutions 'Iieiiave in two din-eta: Ann's depending on Whether the initial value lel is greater than or less than a certain critical value yo. Estimate this critical initial value. yo = -_—_—~-—— (d) Plot the solution that satisﬁes the initial condition mm 2 0‘9. Sketch its graph on the axes below ses the t axis. (I) (6) Estimate the time at which this solution cro 65.24ot) ()7. ()8. ()0 TEST 1 l’7chruzu‘y lo l994 Nulnc Section 07 ()8 ()9 Please answer ull questions, showing your work in detail. You may refer to one sheet (8.5 by l l inches) of notes. No other notes. books, references, calculators, etc. are permitted. Problems 3 and 4 count 25 points each; problem 5 counts 20 points. 'l‘O'l'AL .\'i\i\ll'i ,_ ,__V SECTION 3 (a) l‘iiiil iiH' (‘XPiii'ii sullitiuii ‘1/ (,',)(J') M iiH‘ iiiiliul \'&lill(‘ [imHmn I ,r _ _.r ("2 2 .l/(U) : l (1.17 15 + 4y (1)) Mini {In} suliiiiuii ui' iiH‘ iiiitiu] \';1ill(‘])l'ui)ii‘lil ‘1'!le + By 2 i17+’3. y( l) L .i. The” (iUH‘HHiIH‘ xi su that 1} Running. finite as ‘1' v» 0. NAME ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, .. Slit‘Tlt ).\" 4 (a) Consider the equation (11/ , — »—* ~--i 3"! 5’"! (H ./l .1) t .1) Find the equilibrium solutions. Slit‘lfll several solutions in the ti} plane. If i/(O) : 3. ﬁnd lim y(t). L" (I)) A tank initially (-ontains 201) gallons ol‘ water and 200 grains of salt. \Vatei' eontaining :5 grams per gallon of salt flows into the tank at a rate, of '2 gallons per minute. The mixture in the tank also ﬂows out of the tank at a. rate of ‘2 gallons per minute. Set up an initial value problem whose solution is the amount of salt in the tank at, any time. (You are NOT asked to solve the pt'ul)le111.) Also determine the amount of salt in the tank after a very long time. NAME ________________________________ _- SECTION “uﬁﬁﬁﬁﬂm 5 (21) Find the sulllt‘iull of the initial value [)I‘L)i)i(3111 I/’ -— 1/ Bi/ (l, l/ U) :41 1’ 4)) i . . . ./ (h) “1114‘\Yl‘UllSkiilll H'. nil/(1') Midi/(1') isr‘“. aim] iii/(11') VJHI) i/(J'i 1'(.1') : [(1') ‘i— 2fj(.17), find the Wmnskieiii ii'3()f(1(.l') lelti 1'(.r i . ...
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Exam1Spring94 - LAB TEST 1 65.2400 — 07 08 09 February...

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