Paper 2, Section II, B

Differential Equations | Part IA, 2007

(i) Find the general solution of the difference equation

uk+1+5uk+6uk1=12k+1u_{k+1}+5 u_{k}+6 u_{k-1}=12 k+1

(ii) Find the solution of the equation

yk+1+5yk+6yk1=2ky_{k+1}+5 y_{k}+6 y_{k-1}=2^{k}

that satisfies y0=y1=1y_{0}=y_{1}=1. Hence show that, for any positive integer nn, the quantity 2n26(3)n2^{n}-26(-3)^{n} is divisible by 10.10 .

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