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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 taxis. (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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 Spring '04
 Yoon
 Differential Equations, Equations, Boundary value problem, Stability theory, cients

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