Explain the determaine 110Difference Equations3.6.1Example AConsider the following Sturm-Liouville system:A²yk-1+ Ayk = 0,(3.203)(3.204)Yo = 0,YN+1 = 0.Comparison with equations (3.177) and (3.178) shows that pk–1=1, qk = 0,rk = 1, ao = 1, a1 = 0, aN = 0, and aN+1 = 1. If we let2 - X = 2 cos 0,(3.205)then equation (3.203) becomesYk+1 –(2 cos 0)yk + Yk-1 = 0,(3.206)which has the solutionYk = c1 cos k0 + c2 sin k0,(3.207)where c1 and c2 are arbitrary constants. The first boundary conditions yo = 0gives c1 = 0. The second boundary condition YN+1 = 0 givessin(N +1)0 = 0(3.208)or(N+ 1)0 = na,n = 1, 2, 3, ....(3.209)Thus, the eigenvalues A, are, from equation (3.205), given by the expressionAn = 2 |1 – cosN +1(3.210)= 4 sin?п %3 1,2, 3, ....2(N + 1),Note that there are only N distinct values of n, i.e., n = 1, 2, ....N, since aftern = N the values of the eigenvalues repeat themselves. The N eigenfunctionsassociated with these eigenvalues can be determined from equations (3.207),(3.209), and (3.210); they arekntOn,k= sinn = 1,2, ...., N.(3.211)N+1

Name: Explain the determaine 110Difference Equations3.6.1Example AConsider t
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Description: Explain the determaine 110Difference Equations3.6.1Example AConsider the following Sturm-Liouville system:A²yk-1+ Ayk = 0,(3.203)(3.204)Yo = 0,YN+1 = 0.Comparison with equations (3.177) and (3.178) shows that pk–1=1, qk = 0,rk = 1, ao = 1, a1 = 0, aN