How do you get the general solution?

Question

I get all the steps in the following solution except for the final step to the get the general solution:

also how can I write the maths more clearly in the description?

$a_n = 7a_{n-1} + (-30) \cdot 2^n$

substitute $a_n=c \cdot 2^n$

$c \cdot2^n=7c \cdot2^{n-1} + (-30)\cdot2^n$

$2c=7c-30\cdot2$

$c={30 \cdot 2\over5} = 12$

general solution: $a_n=c \cdot 7^n+12\cdot2^n$

Answer 1

This is an inhomogeneous linear problem. The general solution of such a problem can be described as follows:

1. Find the general solution of the associated homogeneous problem $$a_n=7 a_{n-1}\ .$$ In our simple example this general solution can easily be guessed as $$a_{\rm hom}(n)=C\cdot 7^n\qquad(n\geq0)\ .$$
2. Find a single solution of the given inhomogeneous problem, using the inhomogeneity as a hint. This you have done, and have found the so-called particular solution $$a_{\rm part}(n)=12\cdot 2^n\qquad(n\geq0)\ .$$
3. The general solution of the given problem then can be written as $$a_n=a_{\rm hom}(n)+a_{\rm part}(n)=C\cdot 7^n+12\cdot 2^n\qquad(n\geq0)\ ,$$ with $C$ an arbitrary real constant. (This $C$ has nothing to do with the $c$ you used in 2.)