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?
Best Answer
The flaw is that you are indeed mistaken, in that the equation you present is
In fact, your comment amounts to a proof that the equation in question is not linear, because the sum of solutions need not be a solution. But you can also think in the following terms: a linear homogeneous equation is of the form $Ly=0$ for some linear operator $L$. But the operator you are applying to $y$ is the operator $f\mapsto f\prime\prime - 2\operatorname{id}$, and that subtraction of $2\operatorname{id}$ makes it non-linear, just like the operator on vectors $x\mapsto Ax - b$ is non-linear if $b\not=0$.
Another immediate way of observing your equation is not linear is observing that the zero function is not a solution.