Advanced Calculus fi..

486 Advanced Calculus, Fifth Edition ,$I.T:PROBLEMS1. Show that the linear differential equationis satisfied bywhere- >=y = Acoswt + Bsinot,and determine a solution, in series form, of the differential equation00d2y+dy44b + y = f (t) = x (an cosnt + bn sinnt).n=l2. Show that, granting the correctness of the necessary term-by-term differentiations of series,the functionCOf(x, t) = C[A, cosncr + B, sinnct] sinnx.n=lRwhere the A, and Bn are constants, satisfies the partial differential equationa2f-2 a2fat2 = -, ax2The differential equation is that of the vibrating string, and the series represents the generalsolution when the ends are fixed, rr units apart.3. Show that, granting the correctness of the necessary term-by-term differentiations of series,the functionf (r, 8) = A0 + z (~,r"n=lsatisfies the Laplace equation in polar coordinates:00cos no + B.r" sin no) -As was shown in Section 5.15, this equation describes equilibrium temperature distributions,electrostatic potentials, and velocity potentials. Every function f that is harmonicin a circular domain r c R can be represented by such a series, as will be shown inSection 9.1 1.If two functions f (x) and fi(x) have the same set of Fourier coefficients:are the functions necessarily identical? In other words, is a function uniquely determinedby its Fourier coefficients? The answer is in the affirmative: