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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?

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2 Answers 2

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For a), just insert the product derivatives $$ 0=[ψ\tilde y''+2ψ'\tilde y'+ψ''\tilde y]+p[ψ\tilde y'+ψ'\tilde y]+qψ\tilde y \\ =ψ\tilde y'' + [2ψ'+pψ]\tilde y'+[ψ''+pψ'+qψ]\tilde y $$ so that for the desired normal form you need to solve the first order equation $0=2ψ'+pψ$, $ψ=\exp(-\frac12\int p(z)dz)$.

For b) I take it that you were asking the original question as a method to solve this? If $y_2=ψy_1$, then from the same formula as above it follows that $$ 0=ψ''y_1+[2y_1'+py_1]ψ' $$ Now take the derivative and insert the original ODE $$ 0=ψ'''y_1+ψ''y_1'+[2y_1'+py_1]ψ'' +[\underbrace{-2py_1'-2qy_1}_{=2y_1''}+py_1'+p'y_1]ψ' $$ This now gives a homogeneous linear system in $y_1$ and $y_1'$, $$ \begin{bmatrix} ψ''+pψ'&2ψ'\\ψ'''+pψ''+[p'-2q]ψ'&3ψ''-pψ' \end{bmatrix} \begin{bmatrix} y_1\\y_1' \end{bmatrix} = \begin{bmatrix} 0\\0 \end{bmatrix} $$ As $y_1$ is non-zero, the determinant of the coefficient matrix has to be zero. This gives a differential equation in the quotient $ψ=\frac{y_2}{y_1}$ only.

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How about something bone-headed like this: $$y'(z)=\frac{(c\,f(z)+d\,g(z))(a\,f'(z)+b\,g'(z))-(a\,f(z)+b\,g(z))(c\,f'(z)+d\,g'(z))}{(c\,f(z)+d\,g(z))^2}? $$ Although why you would need to do that is not clear. Maybe you could type up a little more context?

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