13 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-36.

In Problems 1 and 2, use separation of variables to find product solutions u = X(x)Y(y) of the given partial differential equation.

  3. Find a steady-state solution ψ(x) of the boundary-value problem

  4. Give a physical interpretation for the boundary conditions in Problem 3.
  5. At t = 0 a string of unit length is stretched on the positive x-axis. The ends of the string x = 0 and x = 1 are secured on the x-axis for t > 0. Find the displacement u(x, t) if the initial velocity g(x) is as given in FIGURE 13.R.1.
    A string of unit length is plotted on an x, g(x) coordinate plane. The ends of the string x = 0 and x = 1 are secured on the x axis. The center portion of the string from x = 1 over 4 to x = 3 over 4 is evenly displaced vertically upward to a height h indicated on the y axis. The vertical displacement is indicated by 2 vertical dotted lines of length h.

    FIGURE 13.R.1 Initial velocity in Problem 5

  6. The partial differential equation

    is a form of the wave equation when an external vertical force proportional to the square of the horizontal distance from the left end is applied to the string. The string is secured at x = 0 one unit above the x-axis and on the x-axis at x = 1 for t > 0. Find the displacement u(x, t) if the string starts from rest from the initial displacement f(x).

  7. Find the steady-state temperature u(x, y) in the square plate shown in FIGURE 13.R.2.
    A square plate is plotted on an x y coordinate plane such that the bottom edge coincides with the x axis and the left edge coincides with the y axis. The bottom, left, and top edge of the plate are labeled u = 0. The right edge of the plate is labeled u = 50. The right top corner of the plate is labeled (pi, pi).

    FIGURE 13.R.2 Square plate in Problem 7

  8. Find the steady-state temperature u(x, y) in the semi-infinite plate shown in FIGURE 13.R.3.
    A semi-infinite horizontal plate of width pi is plotted on an x y coordinate plane such that the bottom edge coincides with the x axis and the left edge coincides with the y axis. The left edge of the plate is labeled u = 50. A thin strip on the top and the bottom of the horizontal edge of the plate is marked out in barred lines and labeled insulated.

    FIGURE 13.R.3 Semi-infinite plate in Problem 8

  9. Solve Problem 8 if the boundaries y = 0 and y = π are held at temperature zero for all time.
  10. Find the temperature u(x, t) in the infinite plate of width 2L shown in FIGURE 13.R.4 if the initial temperature is u0 throughout. [Hint: u(x, 0) = u0, –L < x < L is an even function of x.]
    A vertically infinite plate plotted on an x y coordinate plane is such that the width 2 L of the plate comes within the interval [negative L, L]. The left and right edges of the plate are indicated by arrows and labeled u = 0.

    FIGURE 13.R.4 Infinite plate in Problem 10

    1. Solve the boundary-value problem

    2. What is the solution of the BVP in part (a) if the initial temperature is

      u(x, 0) = 100 sin 3x – 30 sin 5x?

  12. Solve the boundary-value problem

  13. Find a series solution of the problem

    Do not attempt to evaluate the coefficients in the series.

  14. The concentration c(x, t) of a substance that both diffuses in a medium and is convected by the currents in the medium satisfies the partial differential equation

    where k and h are constants. Solve the PDE subject to

    where c0 is a constant.

  15. Solve the boundary-value problem

    where u0 and u1 are constants.

  16. Solve Laplace’s equation for a rectangular plate subject to the boundary-value conditions

  17. Use the substitution and the result of Problem 16 to solve the boundary-value problem

  18. Solve the boundary-value problem

  19. A rectangular plate is described by the region in the xy-plane defined by In the analysis of the deflection of the plate under a sinusoidal load, the following linear fourth-order partial differential equation is encountered:

    where and D are constants. Find a constant C so that the product

    is a particular solution of the PDE.

  20. If the four edges of the rectangular plate in Problem 19 are simply supported, then show that the given particular solution satisfies the boundary conditions