Second order non-homogeneous recurrence relation

Question

I have a trouble with solving the following equation:

$ a_{n+2} = 5a_{n+1} + 14a_n + (72n + 124) * (-2)^n $, where $a_0 = 8, a_1 = 17$.

I've solved the associated homogeneous equation and got $C_1 * (-2)^n + C_2 * 7^n$ ($C_1$ and $C_2$ are constants). I have no idea what to do after this point.

Could somebody explain it to me, please?

Answer 1

Just like with linear differential equations, you have to find a particular solution of the non-homogeneous recurrence equation, and add it to the general solution of the homogeneous recurrence equation.

Now the non-homogeneous part has the standard form of an exponential times a linear polynomial. So a particular solution we're seeking for will have the form $y_n=p(n)(-2)^n$, where $\deg p$ has degree $1$ more, because $-2$ is a simple root of the characteristic equation: $$y_n=(\alpha n^2+\beta n)(-2)^n$$ Can you proceed now?

Answer 2

Hint

WLOG $a_m=b_m+c_0+c_1m+\cdots$

Replace the values of $a_j, j=n,n+1,n+2$ and compare the coefficients of $n^0,n^1,n^2$

to find $c_j=0\forall j\ge2$ and the values of $c_0,c_1$

so that $$b_{n+2}-5b_{n+1}-14b_n=0$$