I was hoping if someone could point me to a (hopefully) not too technical proof of why the sum of the homogeneous and particular solution to a (linear) differential equation yields the full solution set that to that differential equation.
Intuitively, I understand this as follows (from the answer to this question):
"Suppose I find one particular solution $x_{(p,1)}(t)$ while my friend finds another one $x_{(p,2)}(t)$. Then the difference $x_c(t)=x_{(p,2)}(t)−x_{(p,1)}(t)$ will satisfy the homogeneous equation, since you get $…=f(t)−f(t)=0…$ in the right-hand side when substituting. So two different particular solutions to my ODE can't be arbitrarily different; they can only differ by a solution to the homogeneous ODE."
I am trying to find a way this can be formalized. I am not looking for a proof on how the sum of the homogeneous and particular solutions satisfy the original differential equation - I am trying to see why this is the full solution set.
Thanks!
