Page 325 - Schaum's Outline of Differential Equations

308 AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS [CHAP. 31

Supplementary Problems

31.12. Verify that any function of the form F(x - kt) satisfies the wave equation (31.4).

31.13. Verify that u = tanh (x - kt) satisfies the wave equation.

31.14. If u=f(x-y), show that
12
5
31.15. Verify u(x, t) = (55 + 22x + x ) sin 2t satisfies the PDE 12x u tt - x u xtt = Au^
4
6
31.16. A function u(x, y) is called harmonic if it satisfies Laplace's equation; that is, u xx + u yy = 0. Which of the following
3x
x
y
3x
2
functions are harmonic: (a) 3x + 4y + 1; (b) e cos 3y; (c) e cos 4y; (d) In (x 2 + y ); (e) sin(e ) cos(e )"?
31.17. Find the general solution to u x = cos y if u(x, y) is a function of x and y.
31.18. Find the general solution to u y = cos y if u(x, y) is a function of x and y.
31.19. Find the solution to u y = 3 if u(x, y) is a function of x and y, and u(x, 0) = 4x + 1.
31.20. Find the solution to u x = 2xy + 1 if u(x, y) is a function of x and y, and u(0, y) = cosh y.

31.21. Find the general solution to u^ = 3 if u(x, y) is a function of x and y.

31.22. Find the general solution to u xy = 8xy 3 if u(x, y) is a function of x and y.

31.23. Find the general solution to u xyx = -2 if u(x, y) is a function of x and y.
31.24. Let u(x, t) represent the vertical displacement of string of length n, which is placed on the interval {x/0 < x < n}, at
position x and time t. Assuming proper units for length, times, and the constant k, the wave-equation models the
displacement, u(x, t):

Using the method of separation of variable, solve the equation for the u(x, t), if the boundary conditions
u(0, t) = u(n, t) = 0fort>0 are imposed, with initial displacement u(x, 0) = 5 sin 3x - 6 sin &c, and initial velocity
u t(x, 0) = O f o r O < ^ < ^ .