Linear Non Homogeneous recurrence relation

Question

Find the explicit formula for given recurrence relation: $$a_{n}-7a_{n-1}+10a_{n-2}=2n^{2}+2$$ With the initial conditions $a_0=0,a_1=1$. I just want to know whether the particular solution will be somewhat like $a_{p} = (cn^{2}+bn+a) \, 1^{n}$ or it would be $a_{p} = (cn^{2}+bn+a) \, 2^{n}$. Which one of the above two mentioned particular solutions are correct.

Answer 1

Sketch of a proof.

• Step 1: Solve characteristic equation: $x^2-7x+10=0$, $x=2, x=5$. Hence, the general solution has form $a_n = k_12^n+k_25^n+f_n$ where $f_n$ is a special solution of the recursion.
• Step 2: Find $f_n$, since the RHS is a polynomial of degree 2. Hence, $f_n$ also has this form. Assume $f_n=an^2+bn+c$. Flug $f_n$ into the relation : $f_n-7f_{n-1}+10f_{n-2}=2n^2+2$ to get $a,b,c$.
• Step 3: After $f_n$ is found, we have general solution is $a_n = k_12^n+k_25^n+f_n$. Replace $n=0$ and $n=1$ and solve for $k_1,k_2$ to get the general formula.