Consider an ordinary differential equation (ODE)
$$F(t, y, y') =0$$
in the unknown $y \in C^{\infty}(I, \mathbb{R})$. Suppose that $F$ depends on a smooth function $f \in C^{\infty}(I, \mathbb{R})$, yet there is no explicit dependence on $f'$ or higher order derivatives.
For example, one such differential equation is
$$y' + \tan(f)y = 0.$$
Under such assumptions, I would like to ask the following question.
Question. Suppose that the general solution of the ODE exists for all $f \in C^{\infty}(I, \mathbb{R})$. Can we then conclude that, as a function of $f$, the solution does not depend explicitly on $f'$ or higher order derivatives?
In general, one cannot hope to find an explicit formula for the solution as a function of $f$. So I wonder whether it really makes sense to speak of "explicit dependence".
In general: The solution may depend on $f'$ (although it's kind of an exploit used here).
An Example: Consider $$y' + fy = 0$$ for $$f(x) = \sin(x)$$ with suitable initial conditions. Then as a solution we get $$y(x) = e^{\cos(x)} = e^{f'(x)}$$
So the solution depends on $f'$.