Exam 10: Partial Differential Equations and Fourier Series

Suppose that both ends of a string of length 25 cm are attached to fixed points at height 0. Initially, the string is at rest and has shape 8 sin , where x is the horizontal coordinate along the string with zero at the left end. The speed of wave propagation along the string is 2 cm per sec. Formulate an initial boundary value problem that describes the shape of the string, u(x, t), over time.

Consider the boundary value problem + 9y = 0, y (0) = 3, y ( $\pi$ ) = 0 Which of these statements is true?

Consider the following periodic function with period 4π: f (t) = 4t, -2π < t ≤ 2π f (t + 4π) = f (t) To what value does the Fourier series converge when t = -8π?

Consider the conduction of heat in a rod 30 cm in length whose ends are maintained at 0°C for all time t > 0. Find the expression for the temperature u(x, t) of position x in the rod at time t if the initial temperature distribution is given by u(x, 0) = Assume = 1 in the heat conduction partial differential equation.

Which of the following represents all solutions of the boundary value problem + 36y = 0, y (0) = 1, y ( $\pi$ ) = 1?

Consider the following periodic function with period : F (t) = F = f (t) The Fourier representation has the form Which of these are the coefficients ?

Consider the following periodic function with period 9: where 18 is the period of f (t). What is the coefficient ? Express your answer in exact form involving π. Do not approximate.

The ends of a rod 75 cm in length are connected to reservoirs that maintain the temperature at 11°C at x = 0 and 20°C at x = 75. The initial boundary value problem governing how heat conducts through the rod is as follows:

Consider the following periodic function with period 5: f (t) = f (t + 5) = f (t) To what value does the Fourier series for f (t) converge for t = 5?

Consider the following periodic function with period 4π: f (t) = 5t, -2π < t ≤ 2π f (t + 4π) = f (t) What is the Fourier series representation for f (t)?

Consider the boundary value problem + 25y = 0, y (0) = -2, y = 0Which of these statements is true?

Suppose f (x) is defined by f (x) = 8 on the interval [0, 9]. Consider the function Compute F(-8).

What is the solution of the following initial boundary value problem? 4 = , u(0, t) = 0, u(2, t) = 0, u(x, 0) = sin(4?x)

Consider the boundary value problem + 25y = 0, y (0) = 0, y = 0Which of the following are eigenvalues for this boundary value problem? Select all that apply.

Consider the following periodic function with period 4: f (t) = f (t + 4) = f (t) To what value does the Fourier series for f (t) converge for t = 2?

Consider the following periodic function with period : f (t) = f = f (t) The Fourier representation has the form What is the value of ?

Consider the following periodic function with period 7: f (t) = f (t + 7) = f (t) To what value does the Fourier series for f (t) converge for t = 0?

For which of these partial differential equations can the method of separation of variables be used to reduce it to a pair of ordinary differential equations? Select all that apply.

Which of the following statements are true? Select all that apply.

Which of the following functions is even? Select all that apply.