Explain the determine Example AAssume that we know that one solution of the equationYk+2 – k(k + 1)yk = 0(3.146)LINEAR DIFFERENCE EQUATIONS103(1)is Yk(k – 1)!. We wish to find a second linearly independent solution.From equation (3.146), we see thatIk = -k(k + 1).(3.147)Substitution of this into equations (3.111) and (3.113) gives= (-1)* (k – 1)!.(2)(3.148)Therefore, the general solution to equation (3.146) isYk = [C1 + c2(-1)*](k – 1)!,(3.149)whereC1andC2 are arbitrary constants. Note that the Casoratian isC(k) = (-1)*+1[(k – 1)!²(2k),(3.150)(2)thus showing thatYk* and y are linearly independent. Therefore,k-1C(k) = AQk = A 1| 4i,(3.111)i=1where A is an arbitrary, nonzero constant. Now(1)(2)(2),(1)(2)C(k).(1),,(1)Yk Yk+1Yk Yk+1Yk Yk+1(3.112)(1)„(1).(1)Applying A-1to both sides givesC(k)(1),(1)Yk Yk+1(1)A-Aye(2).(1)-1Yk= YkA(3.113)(1),,(1)Yk Yk+1

Name: Explain the determine Example AAssume that we know that one solution o
Uploaded: 2024-03-20T06:47:08.000Z
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Description: Explain the determine Example AAssume that we know that one solution of the equationYk+2 – k(k + 1)yk = 0(3.146)LINEAR DIFFERENCE EQUATIONS103(1)is Yk(k – 1)!. We wish to find a second linearly independent solution.From equation (3.146), we see thatI