so I can solve some simple difference equations (those with a constant inhomogeneous segment), but I have come across one that I don't know how to deal with. Please see the attached image.
An example of an equation I am having trouble with and one that I can do already.
My biggest issue is how to tackle the (1/6)^n
Any help would be much appreciated!
On
An example of an equation I am having trouble with: $\;\displaystyle x_{n+1}=\frac{1}{6} x_n + \left(\frac{1}{6}\right)^n$
... and one that I can do already: $\;\displaystyle x_{n+1}=\frac{1}{6} x_n + \frac{1}{6}$
Multiply the first equation ("having trouble with") by $6^{n+1}\,$:
$$ 6^{n+1}x_{n+1} = 6^n x_n + 6 $$
Let $\,y_n = 6^n x_n\,$, then the above reduces to:
$$ y_{n+1} = y_n + 6 $$
The latter is of the kind "that I can do already" (it's actually an arithmetic progression in this case). Solve it for $\,y_n\,$, then you get $\displaystyle\,x_n = \frac{1}{6^n} y_n\,$.
The first thing to try with an inhomogeneous term of the form $r^n$ is a constant times $r^n$. But if $r^n$ itself is a solution of the homogeneous equation, the next thing to try is a constant times $n r^n$.