# 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.

• It is called superposition and it applies for linear systems – Dale Nov 9 '18 at 12:30
• The answer (by CuriousKev) has the answer to your question: "Notice your equation of motion is linear in x. So we can take a linear combination of any two solutions and get another one." If you have an eom that is linear, you can add two solutions to get a new, third solution. – Avantgarde Nov 9 '18 at 12:33

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.