Advanced Calculus fi..

690 Advanced Calculus, Fifth Edition4. Prove that if the constants b, are bounded, then the series (10.99) can be written for eacht > 0 as a power series in x, converging for all x. [Hint: Let t > 0 be fixed and let00V(X, y) = bn sin nx cosh ny e-"".".n=lShow that the series for v converges uniformly for -00 < x < oo, -yl 5 y 5 yl by applyingthe M-test withI > ?M M~-?"-C-~n -for n sufficiently large. Show that the series remains uniformly convergent after differentiationany number of times with respect to x and y. Each term of the series is harmonic inx and y; hence conclude that v(x, y) is harmonic for all x and y. By Section 9.1 1, v(x, y)can be expanded as a power series in x and y; put y = 0 to obtain the desired series forU(X, t1.15. Prove that if the constants bn are bounded, then the series (10.99) remains uniformlyconvergent for t 2 tl > 0, -w < x < co, after differentiation any number of times withrespect to x and t.rrrl , 1'6. Prove that if the constants b, are bounded, Ib, I < M, then the function u(x, t) defined by(10.99) converges uniformly to 0 as t + oo; that is, given 6 > 0, a to can be found suchthat lu(x, t)l < E for t > to and -w < x < w. [Hint: Show that lu(x, t)l is less than thesum of the geometric series M ~ (e-rZ')" z , .I7. Let U (X, t) have continuous derivatives through the second order in x and t for t >_ 0 and0 5 x 5 n and let u(0, t) = u(n, t) = 0. Prove that if u(x, t) satisfies the heat equation(10.91) for t > 0,0 < x < rr, then u(x, t) has the form (10.99).[Hint: See Problem 6following Section 10.8.18. Discuss the nature of the solutions for 0 < x i n, t > 0 of the equationwith boundary conditions u(0, t) = u(n, t) = 0, if p, H, and K are positive constants.9. Prove that if A 0, the equations (10.94) and (10.95) have no solution other than thetrivial one: A(x) = 0.We now consider the general problem of type (f):a2u au a2uAx)- + H(x)- - K2- = F(x, t), 0 < x < L, t > 0, (10.111)at2 at ax2This is the problem of response of the 1-dimensional system to outside forces varyingboth in position and time. The fact that the boundary conditions (10.1 12) are variableshows that the motion is being forced at the ends x = 0, x = L also.