Consider the n-th order linear differential equation: u (n) + an−1u (n−1) + . . . + a1u ′ + a0u = 0, (∗) with constant coefficients, whose characteristic polynomial is p(λ) = λ n + an−1λ n−1 + . . . + a1λ + a0 = 0. Suppose that λj ∈ C occurs with multiplicity k as a root of p. Show that e λj t , teλj t , . . . , tk−1 e λj t , are all solutions to (∗). . Consider the n-th order linear differential equationu(n)+ an-1u(n-1)+...+ aju' + aou = 0,(*)with constant coefficients, whose characteristic polynomial isp(X) = X" + an-1A"-1+ a1d+ ao0....Suppose that d; E C occurs with multiplicity k as a root of p. Show thatedst, test,..., ek-1edst,tk-led;t,are all solutions to (*).

As per the question we are given a nth order linear homogeneous differential equation in u(t), Whose…

. Consider the n-th order linear differential equation un) + an-1un-1) +...+a1u' + aou = 0, (*) with constant coefficients, whose characteristic polynomial is p(A) = X" + an-1A"-1 +...+ a1^+ ao = 0. Suppose that d; E C occurs with multiplicity k as a root of p. Show that elst, test,…...,t*-1st, are all solutions to (*).

. Consider the n-th order linear differential equation un) + an-1un-1) +...+a1u' + aou = 0, () with constant coefficients, whose characteristic polynomial is p(A) = X" + an-1A"-1 +...+ a1^+ ao = 0. Suppose that d; E C occurs with multiplicity k as a root of p. Show that elst, test,…...,t-1st, are all solutions to (*).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the n-th order linear differential equation: u (n) + an−1u (n−1) + . . . + a1u ′ + a0u = 0, (∗) with constant coefficients, whose characteristic polynomial is p(λ) = λ n + an−1λ n−1 + . . . + a1λ + a0 = 0. Suppose that λj ∈ C occurs with multiplicity k as a root of p. Show that e λj t , teλj t , . . . , tk−1 e λj t , are all solutions to (∗).