Prelim 2 Math 2930 Spring 2011
show work, 5 problems, no calculators
1) (a) (4 points) Write your name and section number on your exam booklet.
In parts (b) and (c) we consider the function of period 2πwith f(x) = xon
the interval −π
2,π
2, and f(x) = 0 for other points in [−π, π].
(b) (8 points) Find the value f(6).
(c) (8 points) Find the coefficient b3, of sin(3x) in the Fourier series of f.
2) The weight of a 1000 kg mass compresses a spring 9.8 millimeter.
(a) (5 points) Tell why the spring constant k= 106N/m.
(b) (5 points) Write the Newton F=ma equation for frictionless un-
forced oscillations of this system, in terms of the vertical displacement from
equilibrium y(t), and verify that it can be rewritten as y00 + 1000y= 0.
(c) (5 points) Write the general solution to the equation of part (b).
(d) (5 points) Will the mass oscillate slower, or faster, than 29.30 cycles
per second?
3) (20 points) Solve the diff eq x00 + 4x0+ 3x= 5 sin(t) + 7 sin(2t) for x(t).
4) Consider the partial differential equation ut= 5uxx −ux, where uis a
function of xand tand subscripts denote partial derivatives.
(a) (5 points) Trying u(x, t) = X(x)T(t), separate the pde into two ordi-
nary differential equations for Xand T.
(b) (5 points) Solve your equation for T(t).
(c) (5 points) Solve your equation for X(x), but to save time, only do the
case where the separation constant is 0.
(d) (5 points) Find a solution uhaving u(x, 0) = 32ex/5initially.
5a) (10 points) Someone has used the method of separation of variables to
find a list of solutions
cos(n2t) sin(nx)n= 1,2,3, . . .
to the PDE utt +uxxxx = 0. Find a solution u(x, t) having initially u(x, 0) =
20 sin(11x) + 29 sin(30x).
5b) (10 points) (not related to 5a) A function g(x) = 1 + f(x)2where fis
odd and differentiable. Determine whether the derivative g0(x) is even, odd,
or neither.
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