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Unformatted text preview: 65.2405 TEST 1 February 14, 1996 Name Please answer all questions, showing your work in detail and giving reasons where appropriate. This is an open book test. NAME _________________________________ 1. Consider the differential equation 19:;
dt 2 (a) Find the general solution by hand; show your work. Check your answer by dsolve if
you wish. (b) The graph of one solution is a straight line. Find this solution. What initial condition
does it satisfy at t = 0? (c) How do other solutions behave as t —> 00, relative to the straight line solution from
part (b)? N AM E _____________________ (d) From the differential equation itself‘ ﬁnd a condition satisﬁed by points where solutions
have a horizontal tangent line. (e) There is one solution that touches, but does not cross, the t—axis. Find where this
solution touches the t—axis. Also ﬁnd where this solution intersects the y—axis. NAME _________________________________ 2. (a) Solve the initial value problem
y" + 4y' + 3y = 0, y(0) = 1. y/(O) = B where B > 0. Note: If you do this by hand, show your work. If you use Maple,
write down your command(s) as well as the result. (b) Find B so that the solution has a maximum at t = 1/2. Also ﬁnd the corresponding
maximum value of y. (c) Is it possible, by choosing B properly, to make the maximum occur at t = 1? If so,
ﬁnd B that does this. Otherwise, show that it is not possible. NAME _______________..________ 3. (a) Solve the initial value problem
y”+%y’+%§y=0, y(0) = 2, y’(0) =3 Note: If you do this by hand, show your work. If you use Maple, write down your
command(s) as well as the result. (b) Plot 3/ versus t and also y’ versus y for 0 S t S 47r. Sketch these plots on the axes
5 below. (c) On each graph mark the points corresponding to t = 0, t = 7r/2, t = 7r, t = 37r/2, and
t = 27r. NAME _______________________ 4. (a) Write down a ﬁrst order linear differential equation with the property that every
solution approaches 5 as t —+ 30. Then solve your equation and conﬁrm that its
solutions have the required property. (b) Consider the equation dy/dt = (y — 1)2 (y — 4). Find all equilibrium solutions. State
whether each one is asymptotically stable, semi stable, or unstable. (c) Write down an equation whose solutions have the graphs shown below. 3 NAME ________________________________ 5. (a) Consider the equation dy/dt = g2 — t. Draw a direction field on a rectangle such
as 0 S t S 6, —3 g y g 3. Include a few solution curves in your plot also. Sketch
below several solutions and describe in a few words how they seem to behave as t increases. (b) A tank initially contains 50 gallons of water and 100 grams of salt. Water containing
4 grams of salt per gallon ﬂows into the tank at the rate of 3 gallons per minute and the mixture in the tank flows out at the same rate. Let Q(t) be the amount of salt in
the tank at time t. Formulate an initial value problem whose solution is Q(t). What value does Q(t) approach as t > oo? ...
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 Spring '04
 Yoon
 Differential Equations, Equations

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