Exam1Fall95 - 65.2400 TEST 1 Name Section Please answer all...

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Unformatted text preview: 65.2400 TEST 1 September 29, 1995 Name Section Please answer all questions, showing your work in detail. You may use one 8.5 by 11 sheet of notes while taking the examination. No other reference material, books, notes, or calculators are permitted. NAME _______________________________ __ SECTION ____________________ __ 3. (a) (10 points) Find the explicit solution y = $03) of the initial value problem dy_ d3: — y2<1—2x>, yo) = ——. Determine the interval in which the solution is valid. 3. (b) (15 points) Find the general solution of my’ — y = —2a:‘1. Sketch several solutions for x > 0. Find the solution that satisfies y( = 1. Determine Where this solution crosses the x—axis. NAME _____________________________ ___. SECTION ____________________ __ 4. (a) (10 points) Consider the equation 3/ = —y2(y * 3)/2. Determine the equilibrium solutions. Sketch them and several other solutions on the axes below. State whether each equilibrium solution is asymptotically stable, semistable, or unstable. ‘1 4. (b) (15 points) A tank initially contains 100 gallons of water and 50 ounces of salt. Additional water containing two ounces of salt per gallon flows into the tank at a rate of 3 gallons per minute and the well—stirred mixture in the tank flows out at the same rate. Write down an initial value problem whose solution is the amount Q(t) of salt in the tank at any time. How must the problem be modified if the rate of outflow is only two gallons per minute? NAME _______________________________ -- SECTION ____________________ __ 5. (a) (15 points) Find the solution of the initial value problem 2y" + 33/ — 2y = 0, y(0) = 3, y'(0) = B. Then find all values of B for which y —> 0 as t —> oo. 5. (b) (5 points) Find a second order linear homogeneous equation with constant coeffi- cients that has y = 62‘ sin 3t as a solution. LAB TEST 1 65.2400 — 13, 14, 15 September 28, 1995 NAME _______________-_.___ SECTION 1. Consider the differential equation dy 1 y cost sint ___.=_______+_.. dt 22 4 2 (a) Use DEplot to draw a direction field for this equation on the rectangle O S t _<_ 12, ——4 S y S 4. Describe in a few words how the solutions behave as t increases. (b) Plot the solution that passes through the point (0,2). Sketch its graph on the axes below. (c) From a plot estimate the time T at which the graph of the solution in part (b) crosses the t-axis. (d) For the solution in part (b) estimate the coordinates of the first relative minimum point to the) right of t = O. 2.Consider the initial value problem 4g” + 93/ + 2y = 10, y(0) = 2, y’(0) = —4. (a) Use dsolve to solve this problem. Write the solution below. (b) Sketch the graph of the solution on the axes below. fl 7. (c) Find the coordinates of the minimum point. (d) Find the limiting value of the solution as t —> 00. (e) Find the time T at which the solution first comes within 0.1 of its limiting value. ...
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This test prep was uploaded on 04/09/2008 for the course MATH 2400 taught by Professor Yoon during the Spring '04 term at Rensselaer Polytechnic Institute.

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Exam1Fall95 - 65.2400 TEST 1 Name Section Please answer all...

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