For two given germs of the functions $f(z), g(z)$, how to construct a differential equation with solutions of the form $$y(z)=\frac{a\,f(z)+b\,g(z)}{c\,f(z)+d\,g(z)}$$ where $a, b, c, d$ are arbitrary complex numbers?
Edit 1: This is the third part of a problem in my ODE class, but I don't know what knowledge I should use to solve this problem. Other parts of the problem are the followings:
For a differential equation with meromorphic coefficients of the form $$y’’+ p(z) y’ + q(z) y = 0$$ a.Find the substitution $y = ψ\widetilde{y}$ that reduces the equation to $$\widetilde{y}’’ + Q(z) \widetilde{y} = 0$$ Where is this replacement defined?
b.Build a differential equation that everyone satisfies functions of the form $y_1 / y_2$, where $y_1, y_2$ are any two nonzero solutions of the equation of the original equation.
Edit 2: Does this have something to do with Möbius transformation?