Answer to Question #75093 in Quantitative Methods for AJIT SINGH

Question

show that ux =c1eax +c2e-ax is the solution of the difference equation ux +1-2 ux Cosha+ux-1 =0

Answer 1

u(x+1)+u(x-1)=c_1 e^a(x+1) +c_2 e^(-a(x+1) )+c_1 e^a(x-1) +c_2 e^(-a(x-1) )=(c_1 e^a(x+1) ++c_1 e^a(x-1) )+(c_2 e^(-a(x+1) )+c_2 e^(-a(x-1) ) )=(e^a+e^(-a) ) c_1 e^ax+(e^(-a)+e^a ) c_2 e^(-ax)=(e^a+e^(-a) )(c_1 e^ax+c_2 e^(-ax) )=2 cosh⁡(a)u(x).

Answer:
Yes, this equation holds for all x∈R.