If I'm not mistaken, the set of all functions $f(x)$ satisfying the first order homogeneous ODE:
$$f''(x) - 2x = 0$$
is a Vector Space (as in, the elements of the Vector Space are its solutions).
Two solutions for the above ODE are $f(x) = x^2 + 7$ and $f(x) = x^2 + 9$.
Therefore, if they are elements of the Vector Space, a linear combination of them say: $2x^2 + 16$, should also be a solution to the ODE above. However, it is not.
Where is the flaw above?