Paper 2, Section I, ADifferential EquationsPart IA, 2010Find the general solutions to the following difference equations for yn,n∈Ny_{n}, n \in \mathbb{N}yn,n∈N. (i) yn+3−3yn+1+2yn=0, (ii) yn+3−3yn+1+2yn=2n (iii) yn+3−3yn+1+2yn=(−2)n (iv) yn+3−3yn+1+2yn=(−2)n+2n.\begin{aligned} \text { (i) } & y_{n+3}-3 y_{n+1}+2 y_{n}=0, \\ \text { (ii) } & y_{n+3}-3 y_{n+1}+2 y_{n}=2^{n} \\ \text { (iii) } & y_{n+3}-3 y_{n+1}+2 y_{n}=(-2)^{n} \\ \text { (iv) } & y_{n+3}-3 y_{n+1}+2 y_{n}=(-2)^{n}+2^{n} . \end{aligned} (i) (ii) (iii) (iv) yn+3−3yn+1+2yn=0,yn+3−3yn+1+2yn=2nyn+3−3yn+1+2yn=(−2)nyn+3−3yn+1+2yn=(−2)n+2n.