Question

2 (a) Use the method of separation of variables to derive a solution of the following initial-boundary-value problem: utt(x, t) - c^2 uxx(x, t) = 0, 0 < x < L, 0 < t, u(0, t) = u(L, t) = 0, 0 ? t, u(x, 0) = ?(x), ut(x, 0) = ?(x), 0 ? x ? L, where L, c are given positive constants and ?, ? are given functions. (b) Find the solution u(x, t) of part (a) when ?(x) = 5 sin(3?/L x), ?(x) = 0, 0 ? x ? L. (c) Find the solution u(x, t) of part (a) when ?(x) = 0, ?(x) = 2 sin(4?/L x), 0 ? x ? L. (d) Use your solutions to parts (b) and (c) to write down the solution u(x, t) of part (a) when ?(x) = 5 sin(3?/L x), ?(x) = 2 sin(4?/L x), 0 ? x ? L. (e) Let u(x, t) be the solution of part (b) and recall that the formula for u(x, t) models wave propagation on a string of length L. For what values of x0 ? [0, L] do we have that u(x0, t) = 0 for all values of t ? 0 ? What is the significance of these values of x0 from the point of view of the waves on the string?

          2 (a) Use the method of separation of variables to derive a solution of the following initial-boundary-value problem:

utt(x, t) - c^2 uxx(x, t) = 0, 0 < x < L, 0 < t,
u(0, t) = u(L, t) = 0, 0 ? t,
u(x, 0) = ?(x), ut(x, 0) = ?(x), 0 ? x ? L,

where L, c are given positive constants and ?, ? are given functions.
(b) Find the solution u(x, t) of part (a) when

?(x) = 5 sin(3?/L x), ?(x) = 0, 0 ? x ? L.

(c) Find the solution u(x, t) of part (a) when

?(x) = 0, ?(x) = 2 sin(4?/L x), 0 ? x ? L.

(d) Use your solutions to parts (b) and (c) to write down the solution u(x, t) of part (a) when

?(x) = 5 sin(3?/L x), ?(x) = 2 sin(4?/L x), 0 ? x ? L.

(e) Let u(x, t) be the solution of part (b) and recall that the formula for u(x, t) models wave propagation on a string of length L. For what values of x0 ? [0, L] do we have that u(x0, t) = 0 for all values of t ? 0 ? What is the significance of these values of x0 from the point of view of the waves on the string?

2 (a) Use the method of separation of variables to derive a solution of the following initial-boundary-value problem:

utt(x, t) - c^2 uxx(x, t) = 0, 0 < x < L, 0 < t,
u(0, t) = u(L, t) = 0, 0 ? t,
u(x, 0) = ?(x), ut(x, 0) = ?(x), 0 ? x ? L,

where L, c are given positive constants and ?, ? are given functions.
(b) Find the solution u(x, t) of part (a) when

?(x) = 5 sin(3?/L x), ?(x) = 0, 0 ? x ? L.

(c) Find the solution u(x, t) of part (a) when

?(x) = 0, ?(x) = 2 sin(4?/L x), 0 ? x ? L.

(d) Use your solutions to parts (b) and (c) to write down the solution u(x, t) of part (a) when

?(x) = 5 sin(3?/L x), ?(x) = 2 sin(4?/L x), 0 ? x ? L.

(e) Let u(x, t) be the solution of part (b) and recall that the formula for u(x, t) models wave propagation on a string of length L. For what values of x0 ? [0, L] do we have that u(x0, t) = 0 for all values of t ? 0 ? What is the significance of these values of x0 from the point of view of the waves on the string?

Added by Alexis G.

Question

Please give Ace some feedback