Paper 2, Section I, A

Differential Equations | Part IA, 2011

(a) Consider the homogeneous kk th-order difference equation

akyn+k+ak1yn+k1++a2yn+2+a1yn+1+a0yn=0a_{k} y_{n+k}+a_{k-1} y_{n+k-1}+\ldots+a_{2} y_{n+2}+a_{1} y_{n+1}+a_{0} y_{n}=0

where the coefficients ak,,a0a_{k}, \ldots, a_{0} are constants. Show that for λ0\lambda \neq 0 the sequence yn=λny_{n}=\lambda^{n} is a solution if and only if p(λ)=0p(\lambda)=0, where

p(λ)=akλk+ak1λk1++a2λ2+a1λ+a0p(\lambda)=a_{k} \lambda^{k}+a_{k-1} \lambda^{k-1}+\ldots+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}

State the general solution of ()(*) if k=3k=3 and p(λ)=(λμ)3p(\lambda)=(\lambda-\mu)^{3} for some constant μ\mu.

(b) Find an inhomogeneous difference equation that has the general solution

yn=a2nn,aRy_{n}=a 2^{n}-n, \quad a \in \mathbb{R}

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