Show that a solution is the general solution.

Question

Find general solution of recurrence relation $$ ax_{n+1}+bx_n+cx_{n-1}=0 $$ for two distinct roots $\alpha$ and $\beta$..

My question is: One solution is $y_n=A\alpha^n+B\beta^n$. But how does one show that this is the general solution?

Answer 1

It is basic linear algebra. The set of solutions (functions from the non-negative integers to (say) the reals) of the recurrence is a $2$-dimensional vector space.

It can be shown that $y_n=\alpha^n$ and $y_n=\beta^n$ are linearly independent solutions, and therefore they span the space of solutions.

Answer 2

Let $E_{R_2}$ the set of solutions of the recurrence relation. It's easy to prove that this set is subspace of $\Bbb R^{\Bbb N}$ so let

$$\Phi : E_{R_2}\to \Bbb R^2,\quad (u_n)\mapsto (u_0,u_1)$$

then we see easily that $\Phi$ is an isomorphism of vector spaces hence $$\dim E_{R_2}=2$$