Cornell MATH2930 Sp2009 Final

8/11/2019 Cornell MATH2930 Sp2009 Final

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Final Exam Math 2930 Spring 2009

show work, 14 problems, no calculators

1) (4 points) Write your name and section number at the top of this page.

2) (16 points) Let f(t) = sint2

when|t|< , andfhas period 2. Sketch

the graph offon the interval [, 4] and work out the Fourier series off.[Do simplify any trig functions of multiples of .]

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3) (10 points) A population p(t) [hundred crocodiles] is modeled by the

equation dpdt = 1 p29 . Sketch a slope field (including negative values ofp),

and decide from your sketch which of the two equilibrium solutions is stable.

4) (10 points) Herey (x, t) is a sum of traveling waves, y(x, t) =f(x + ct) +g(xct).

a) To achieve the boundary condition y(6, t) = 0 for all t, show that thefunctionsf andg must be related by g(s) =f(12 s) for every number s.

b) Give an example of an even function f of period 12. Using your f andthe result of part a), show that you automatically get the initial conditiony(x, 0) = 0.

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5)(10 points)For the crocodiles in Problem 3, we define a new function v(t)

by setting p(t) = 3 + 1v(t) . Show thatv must be a solution to a linear firstorder equation and solve that to find v .

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6) (10 points) A mass m = 10000 kg hangs at the end of a steel cable

and the weight stretches the cable 0.01 meter, like a spring. The weight ismg [Newton] where g = 9.8. A motor running nearby also exerts a forcecos(t) + sin(t) which causes the weight to vibrate up and down a little.As for most electric motors, is about 300. The displacement y(t) of theload from equilibrium is described by

my +ky = cos(t) + sin(t)

Find the spring constantk [Newton/m] for the cable, work out at least onesolution to the differential equation in the typical nonresonant case, andexplain why resonance does not occur.

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8) (10 points) Find all eigenvalues and a corresponding eigenfunction for

each, in the boundary value problem

2y +y = 0

y(0) = 0

y(2) = 0

You may assume 0.

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9) (10 points) A string vibrating in air friction is described by the problem

wtt+ 1

10wt =

1

75wxx

w(0, t) = 0

w(, t) = 0

Somebody wants to separate this problem using w(x, t) = X(x)T(t). De-rive the ordinary differential equations for X(x) and T(t), including whathappens to the boundary conditions. You are not asked to solve anything.

10) (10 points)Someone has worked out a recursion (n + 2)2

cn+2= cn forcoefficients in a power series

y(x) =c0+c1x+c2x2 +c3x

3 +

Findc0 throughc5 for the case where y(0) = 0 and y(0) = 1.

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11)(10 points)A rectangular tank measures 2 meters east-west by 3 meters

north-south and contains water of depthx(t) meters, wheret is measured inseconds. One pump pours water in at the rate of 0.05 [m3/sec] and a secondvariable pump draws water out at the rate of 0.07 + 0.02 cos(t) [m3/sec].The variable pump has period 1 hour.

Set up a differential equation for the water depth, including the correct valueof . Do not solve it.

12) (10 points) This problem concerns the Euler numerical method for thecrocodile equation of Problem 3, with stepsize h, and initial value p(0) = 0.

Leave h as a symbolic variable. Write out expressions for p1, p2, p3 whichare the Euler approximations forp(h),p(2h), andp(3h). [Dont simplifyp3.]

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13) (10 points) Solve the equation

dy

dx= (x+ 3)2

with the initial value y (0) =2.

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14) (10 points) Find a solution to the heat problem

ut = 1

200uxx

u(0, t) = 0

u(, t) = 0

u(x, 0) = 5 sin(x) +1

3sin(2x)

The solution to this problem might be on your formula sheet, so you mustEither: derive your solution by the separation of variables method,Or: write out a clear verification that your solution does in fact satisfy all

the conditions.

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Cornell MATH2930 Sp2009 Final

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