Page 397 - Schaum's Outline of Differential Equations

380 ANSWERS TO SUPPLEMENTARY PROBLEMS

CHAPTER 31

31.16. (a) harmonic; (b) harmonic; (c) not harmonic; (d) harmonic; (e) not harmonic

31.17. x cos y +f(y), where f(y) is any differentiable function of y

31.18. sin y +f(x), where f(x) is any differentiable function of x

31.19. 3^ + 4^+1
2
31.20. x y + x + cosh y
31.21. — x 2 + xg(y) + h(y), where g(y) and h(y) are any differentiable functions of y

2 4
31.22. u(x, y) = x y + g(x) + h(y), where g(x) is a differentiable function of x, h(y) is a differentiable function of y
2
31.23. u(x, y) = — x y + g(x) + xh(y), where g(x) is a differentiable function of x, and h(y) is a differentiable function of y
31.24. u(x, t) = 5 sin 3x cos 3kt - 6 sin 8x cos 8kt

CHAPTER 32

32.22. y = 0

32.24. y = sin x

32.26. y = B cos x, B arbitrary 32.27. No solution

32.28. No solution 32.29. y = x + B cos x, B arbitrary

32.30. 'k=l, y = c le- x 32.31. No eigenvalues or eigenfunctions

32.33. A,= 1, y = c 2e x (c 2 arbitrary)

2
32.34. X n = -n jf, y n = A n sin rim (n = 1, 2, ...) (A n arbitrary)

2
32.36. X n = n , y n = B n cos nx (n = 1, 2, ...) (B n arbitrary)
32.37. Yes 32.38. No, p(x) = sin TK is zero at x = + 1, 0.

32.39. No, p(x) = sin x is zero at x = 0. 32.40. Yes

32.41. No, the equation is not equivalent to (29.6). 32.42. No, w(x) = s not continuous at x = 0.