HARI SIR, a maths doubt

Instead of generating functions, I will give you an algorithmic approach known as annihilators which is derived from Gen Funcs and notions of continuity

We first adapt the equation into a sequence form and write it as a_{n+2}-\sqrt 2 a_{n+1}+a_n = 0

Derive the characteristic polynomial x^2-\sqrt 2 x+1 = 0

which has the roots e^{i \frac{\pi}{4}}, e^{-i \frac{\pi}{4}}

From this derive the sequence as A \left(e^{i \frac{\pi}{4}} \right)^n+ B \left(e^{-i \frac{\pi}{4}}\right)^n = Ae^{i n\frac{\pi}{4}} +Be^{-i \frac{n\pi}{4}}

Usually we will be given a₀ and a₁ to get at A and B

But here we dont really need them to arrive at the conclusion that f(x+8) = f(x)

If you let A = B, and extend using continuity etc. you get f(x) = A cos πx/4 as a function satisfying the given equation

Sir, after finding the roots, what is the purpose of writing the equation containing A & B with the roots?
Moreover what is the relation of a₀ ,a₁ with A & B? How can we find A & B with the help of a0 & a1?
Why had u taken A=B????? It was not given.

Sir, these doubts still persists. Pls clarify.

and sir, why had u taken a_n,a_n+1,a_n+2 as equal......i.e, equal to x ??

?????????