1. A horizontally stretched string of length L = 7 is subject to a nonuniform body force per unit

mass Q(x, t), so that the resulting vertical displacement u(x, t) satisfies

J²u

21²

and the initial conditions are

J²u

əx²

(1)

where e is a constant. The ends of the string are looped around and free to slide on vertical

frictionless wires so that the boundary conditions are

+ Q(x, t)

Ju(0, 1) — 0, Ju(π, l)

Әх

ər

u(x,0) = 0,

Ju

ƏL

=

0.

(x,0) = Vo

where Vo is a constant.

(a) In (1), what are the dimensions of c? What does e represent physically?

Next, seek a solution to (1-3) in the form

u(x, t) = Σan (1)on(x)

n=0

(2)

(3)

(4)

by following the steps below:

(b) Assume a separable solution to the homogeneous version of the PDE and boundary con-

ditions (1)-(2) of the form u = o(r)h(t). Write down or find the eigenvalues An and

eigenfunctions on(r) of the homogeneous problem.

(c) Using the eigenfunction expansion (4) and your results from (b), find the second-order

ODEs satisfied by each unknown time-dependent amplitude an (1) for Q(x, t) in (1) given

by

Q(x, t) = cos + et cos(2x)

(5)

where is a constant.

(d) Find the initial conditions for an (l) for all n = 0, 1, 2....

(e) Using the ODEs and initial conditions found in (c) and (d), obtain the general solution

for an (1) for all n = 0, 1, 2....

(f) Write the final expression for u(x, t).