In this problem we indicate one way to show that if r = r1

Chapter 4, Problem 41

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In this problem we indicate one way to show that if r = r1 is a root of multiplicity sof the characteristic polynomial Z(r), then er1t, ter1t, ... , ts1er1t are solutions of Eq. (1).This problem extends to nth order equations the method for second order equations givenin of Section 3.4. We start from Eq. (2) in the textL[ert] = ertZ(r) (i)and differentiate repeatedly with respect to r, setting r = r1 after each differentiation.(a) Observe that if r1 is a root of multiplicity s, then Z(r) = (r r1)sq(r), where q(r) isa polynomial of degree n s and q(r1) = 0. Show that Z(r1), Z(r1), ... , Z(s1)(r1) are allzero, but Z(s)(r1) = 0.(b) By differentiating Eq. (i) repeatedly with respect to r, show thatrL[ert] = Lrert= L[tert],...s1rs1 L[ert] = L[ts1ert].(c) Show that er1t, ter1t, ... , ts1er1t are solutions of Eq. (1).