Suppose we know all functions $\phi$ that are solutions of the differential equation
for some arbitrary function $F_1$. Clearly those same $\phi$ also satisfy the differential equation
obtained by "differentiating the differential equation". Intuitively, I assume new solutions are generated, but I'm not certain if that is always the case and if there is a systematic way to relate those new solutions to the ones we already have for the original differential equation and to the functional form of $F_1$ and $F_2$. For example, the solutions to the differential equation
are of the form $y(x)=C$ for some constant $C$. "Differentiating" the differential equation, we get
which has solutions of the form $y(x)=C_1x+C_2$ for some constants $C_1$ and $C_2$. The set of solutions of the second differential equation contains the solutions of the first one, as expected, but also contains new solutions, which in this case are antiderivatives of the original solutions. It doesn't seem that things work out always as they do for this simple case, though.
Is there a systematic way of knowing what will happen to the solution set of a differential equation if you "differentiate" it?