tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 1

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 4 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Homogeneous Constant Coefficient Equations Homogeneous constant coefficient equations have the form an N PN-ian N-1 ---- P1 n 1 poan 0. The substitution an rn yields rN PN-irN-1 --- Pir P0 0 r r1 m1 r rk mk 0. If r1 is a distinct root then the associated linearly independent solution is n- If r1 is a root of multiplicity m 1 then the associated solutions are rn nrn n2rn . nm 1 r l . Result Consider the homogeneous constant coefficient difference equation an N pN 1 an N 1 p1 an 1 p0an 0. The substitution an rn yields the equation r r1 mi r rk mk 0. A set of linearly independent solutions is ir nr nmi 1rn r nr nmk Cnl V 1 n 1 . n 1 . k n k . . . n k J . Example Consider the equation an 2 3an 1 2an 0 with the initial conditions a1 1 and a2 3. The substitution an rn yields r2 3r 2 r 1 r 2 0. Thus the general solution is an C11n C22n. 1174 The initial conditions give the two equations ai 1 Ci 2C2 a2 3 Ci 4c2 Since C1 1 and c2 1 the solution to the difference equation subject to the initial conditions is an 2n 1. Example Consider the gambler s ruin problem that was introduced in Example . The equation for the probability of the gambler s ruin at n dollars is an pan i qan-i subject to a0 1 aN 0. We assume that 0 p 1. With the substitution an rn we obtain r pr2 q. The roots of this equation are 1 1 4pq r ------------- 2p 1 V1 4p 1 p 2p 1 ự 1 2p 2 2p _ 1 1 2p 2p . We will consider the two cases p 1 2 and p 1 2. .