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?
Best Answer
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?