Adding solutions to get a new solution? Recently I've stumbled upon the same question as here: What happens to the position function when an oscillator is overdamped and does not have angular frequency?
And the answer that I preferred (the one by CuriousKev) was perfect until the last sentence, "Since we can add solutions to get a new solution, the most general is:
X = A−r1t + B-r2t "
I've tried searching for the "adding two solutions to get final solution" theory but I either get mistaken for chemistry questions or other unrelated mathematical questions, so could someone link me to the right place or explain this usage of solution(s) addition?

I could be posting the question in the wrong place (as in maybe I should ask such question on maths.stackexchange instead?) It is my first question around this website; so apologies in advance.
 A: Great question! This is simply a feature of linear differential equations. If I have a differential equation with 2 solutions $y_1$ and $y_2$, then $a y_1 + b y_2$ must be a solution. I'll prove that here quickly:
Let's say we have the differential equation
$$y'' + \beta^2 y' = 0$$
If $y_1$ is a solution, then:
$$y_1'' + \beta^2 y_1 = 0$$
and if $y_2$ is a solution, then:
$$y_2'' + \beta^2 y_2 = 0$$
Now, let's say I want to check if $Y = a y_1 + b y_2$ is a solution. If it is, then the following must equal 0:
$$
Y'' + \beta^2 Y \\
= (a y_1 + b y_2)'' + \beta^2 (a y_1 + b y_2) \\
= (a y_1)'' + (b y_2)'' + \beta^2 a y_1 + \beta^2 b y_2 \\
= a y_1'' + b y_2'' + \beta^2 a y_1 + \beta^2 b y_2 \\
= a (y_1'' + \beta^2 y_1) + b (y_2'' + \beta^2 y_2) \\
= a(0) + b(0) \\
= 0
$$
Essentially, A linear combination of any number of solutions to a linear differential equation must be a solution to that differential equation
Hope that helps!
EDIT: Wanted to clarify that this is necessarily true only for linear differential equations.
