CHAP. 12]
PARTIAL DIFFERENTIAL EQUATIONS
279
12.42. Find general solutions of each of the following. ( A ) (B) (o)
12.43. Find general solutions of each of the following. (A)
(E)
(6)
(d)
12.44. Solve 12.45. Show that the general solution of
SEPARATION OF VARIABLES 12.46. Solve each of the following boundary-value problems by the method of separation of variables.
(
a
( ( (d)
)
0
b c
) )
z
u(x, 0 ) = 40-*
,
u(x, 0 ) = Ze~5x + 2e-3*
, t
(
0
, t) = 0,
ux(0,t) = 0,
u(v, t) = 0, u(2,t) = 0,
u(x, 0) = 2 sin Zx — 4 sin 5x w(*,0) = 8cos^-6cos^
12.47. Solve the boundary-value problem y(0,t) = y(5,t) - 0, if
(a) f(x) — 5 sin wx,
y(x,0) = 0,
yt(x,Q) = /(»)
(b) f(x) = 3 sin Zvx — 2 sin 5irx.
SOLUTIONS USING FOURIER SERIES 12.48. (a) Solve the boundary-value problem «(0,t) = M(4,<) = 0,
M(*,0) - 25«
where 0 < x < 4, t > 0. (6) Interpret physically the boundary-value problem in (a). 12.49. (a) Show that the solution of the boundary-value problem ux(0, t) = ux(ir,t) - 0, where 0 < x < ir, t>0 is given by
(b) Interpret physically the boundary-value problem in (a).
u(x,Q) - f(x)