Find a complete solution to the difference equation $y_{n+2} + 2y_{n+1} - 3y_n = 2 + 2n$

Question

Find a complete solution to the difference equation $$y_{n+2} + 2y_{n+1} - 3y_n = 2 + 2^n$$

Answer 1

Hint

I am sure that you already solve the case of

$$y_{n+2} + 2y_{n+1} - 3y_n =0$$

Now, since this is a linear recurrence equation, if $$y_{n+2} + 2y_{n+1} - 3y_n = P_k(n)$$ where $P_k(n)$ is a polynomial of degree $k$, the particular solution must be a polynomial $Q_{k+1}(n)$ that is to say $$y_n=Q_{k+1}(n)=\sum_{i=0}^{k+1} a_i\, n^i$$ Just replace and identify the coefficients $a_i$.