When solving a secondary differential equation, how do you know the solution is as general as possible?

Question

However how do you know that having y= C.F + P.I gives you the most general solution to the diff. equation, can you prove such a thing?

Answer 1

I'm not exactly sure what you're asking, but I'm assuming you're asking when we know we have the most general answer to a second order ODE. Well, the answer is that every $n$th order homogeneous ordinary differential equation will have exactly $n$ linearly independent solutions. (Discusssion on this topic) The inhomogeneous counterparts to these homogeneous solutions will have the same linearly independent solutions plus a particular solution specific to the problem. In short, for a second order ODE, if you have an equation of the form $$f(x,y,y',y'')=g(x) \ \mathbf{(1)}$$ Where the homogeneous solution, $y_H$ satisfies $$f(x,y,y',y'')=0$$ and is of the form $$y_H(x)=c_1y_1(x)+c_2y_2(x)$$ Where $y_1$ and $y_2$ are linearly independent functions, then the general solution is given by $$y_G(x)=y_H(x)+y_P(x)$$ where $y_P$ is some particular solution to (1). This will encapsulate every possible solution of (1).