How to solve the particular solution to a recurrence relation?

Question

Here is my solution to a problem:https://i.stack.imgur.com/9eH7O.jpg, though the answer should be:n+2/3 for the particular solution.

Where am I going wrong?

P.S. I am focusing on the particular solution right now. As that is where I am stuck.

Answer 1

I don't understand why are you taking it into differential equation.

Directly put $a_n=An+B$ into the recurrence relation.

$$A(n+2)+B-10(An+A+B)+21(An+B)=12n$$ $$\Longrightarrow 12An-8A+12B=12n$$

Comparing coefficients on both sides, we get,

$$A=1, B=\frac{2}{3}$$

as required.

Hope it helps:)