Take a linear, non-homogenous ODE:
$$ Ay^{\prime\prime}(x) + By^\prime(x) + Cy(x) = f(x) $$
It is known that one can find the general solution to this equation by taking a solution to the following:
$$ Ay_h^{\prime\prime}(x) + By_h^\prime(x) + Cy_h(x) = 0 $$
And a particular solution of the original ODE:
$$ Ay_p^{\prime\prime}(x) + By_p^\prime(x) + Cy_p(x) = f(x) $$
By adding the two together, you get the following:
$$ A\Big(y_p^{\prime\prime}(x) + y_c^{\prime\prime}(x)\Big) + B\Big(y_p^\prime(x) + y_c^\prime(x)\Big) + C\Big(y_p(x) + y_c(x)\Big) = f(x) $$
Thus, it is proven that a solution to the ODE is given by $y_p(x) + y_c(x)$. Why, though, is this the most general solution? I'm aware the solution is made more general by the introduction of the general homogenous solution, but how do we know that there isn't a more comprehensive solution?