Question

Find the solution of the difference equation y_(k+2)-4y_(k+1)+4y_k=0;k=0,1,2,...
Also find a particular solution when y_0=1 and y_1=6.

Answer 1

(i) The characteristic equation is $C^2-4C+4=0$.

The roots are $C_{1,2}=2≔C$.

The general solution of the difference equation is $y_k=A_1C^k+A_2kC^k=A_12^k+A_2k2^k$.

(ii) From the boundary conditions, we get that $A_1=1$, $2A_1+2A_2=6$.

Answer:

(i) $y_k=A_12^k+A_2k2^k$.

(ii) $A_1=1$, $A_2=2$.

References:

1 $\url{https://en.wikipedia.org/wiki/Recurrence_relation#Solving_homogeneous_linear_recurrence_relations_with_constant_coefficients}$