1) Consider the function f(t) defined over one half-period 0 £ t £ L:   3t    if 0 £ t < 1  3    if 1 £ t < 2  a    if t = 2       F(t) =   Where a is an unknown constant

1) Consider the function f(t) defined over one half-period 0 £ t £ L:

Where a is an unknown constant.

1(a) Let f(t) be the odd extension of f(t). Find the value of the constant a that ensures the Fourier Sine Series of f(t) converges to the odd extension fo(t).

1(b) Sketch a plot of f(t), the odd extension of f(t). Make sure you label your axes and cover the interval (-L, L). Be sure to plot the value of a that you found in part (a).

1(c) Calculate the Fourier Sine Series of f(t).

2) Consider the mass-on-a-spring problem described by the following ODE:

3x” + 12x = F(t)

Where F(t) is a periodic function with the Fourier Series:

               F(t) = S_{n odd}4/n sin(nt)

2(a) Find the solution x(t) to the ODE. For full credit, you must derive your solution. Do not use a formula from the textbook.

2(b) What is the natural frequency w of the mass-on-s-spring-system? With this forcing function F(t) given in part (a), do you expect resonance to occur? Explain why or why not in least one complete sentence.

3. Consider the following boundary value problem for a vibrating string:

            y_tt = 25y_xx,       0 < x < 2,   t > 0

      y(0, t) = y(2, t) = 0

      y(x, 0) = 1 – 3 cos(3px)

     y_t(x, 0) = 7 sin(2px)

The solution to the wave equation is:

Y(x, t) = S^¥_n=1A_n cos(npat/L) sin(npx/L) + S^¥_n=1B_nsin(npat/L) sin(npx/L)

Find y(x, t).

4. Consider the Dirichlet problem for Laplace’s equation on the semi-infinite strip:

u_xx + u_yy = 0      (0 < x < a,  y > 0)

u_x(0, y) = u_x(a, y) = 0

u(x, 0) = f(x)

u(x, y) is bounded as    y ® +¥

4(a) Assume that the solution u(x, t) can be represented as a product of two single-variable functions.

u(x, y) = X(x) Y(y)

Use Separation of Variables to derive two ODEs: one for X(x) and one for Y(y). Be sure to include the relevant boundary conditions for each ODE.

4(b) Solve the ODEs that you derived in part (a). Make sure that your solutions satisfy the relevant boundary conditions.

4(c) Derive the full solution for u(x, t) using the results you found in part (a) and part (b). Make sure that your solution satisfies all the conditions in the Boundary Value Problem in Equation (1).