I would like to know if the set of solutions of the following third order linear ODE is convex.
$$y''' + y'' -2y' + y = e^x $$
Can I do that by solving it? Can I infer that from the fact that $y(x) = e^x$ is a particular solution and is a convex function?
The solutions $y$ to any linear differential equation $L\,y=f$, where $L$ is a linear differential operator and $f$ is a source function, form a convex set of functions. To show this, suppose $y_1$ and $y_2$ are solutions, and $k_1$ and $k_2$ are any scalars, then $$L\,\left(k_1\,y_1+k_2\,y_2\right)=k_1\,\left(L\,y_1\right)+k_2\,\left(L\,y_2\right)=k_1\,f+k_2\,f=\left(k_1+k_2\right)\,f$$ by linearity of $L$. Now, if $k_1+k_2=1$, then we conclude that $$L\,\left(k_1\,y_1+k_2\,y_2\right)=f\,,$$ whence $k_1\,y_1+k_2\,y_2$ is also a solution. This shows that the solution set is therefore convex (in fact, affine, as remarked by Giuseppe Negro). Thus, all you have to check is whether your differential operator $L$ is linear, and in your problem, $L$ is indeed linear. More generally, the solution set of any linear equation (not necessarily a differential equation) is an affine (whence convex) set.