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I have a function $f(x_1, \ldots , x_n)$ in $n$-variables that satisfies a fourth order linear differential equation (say $Df=0$). The function $f(x_1, \ldots, x_n)$ is of the form $f(x_1, \ldots, x_n)= g(x_,\ldots, x_n) h(x_1, \ldots, x_n)$ where $g$ is a known analytic function and $h$ is an unknown function that should be solved for. Is there a way to use mathematica to plug in $f=gh$ in the differential equation and get a differential equation in just $h(x_1, \ldots, x_n)$? In other words I want to get some differential equation $D'h=0$. For example, take the case $n=2$ and

$$D= \frac{\partial^4}{\partial x_1^4}-\frac {1}{{x_1-x_2}}\frac{\partial}{\partial x_1 \partial x_2}+\frac{1}{{(x_1-x_2)^2}}\frac{\partial }{\partial x_2},$$ and $g(x_1,x_2) = \frac{1}{(x_1 x_2)^{1/4}(\sqrt{x_1}+\sqrt{x_2})^{1/2}}$ and we know that $D(gh)=0$ then is there a way to find a differential equation $D'h=0$?

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  • $\begingroup$ What is the question here? $\endgroup$ – Alex Trounev Oct 2 '19 at 19:19
  • $\begingroup$ Well, I want to do the above mentioned simplification using mathematica instead by hand. My $g$ is quite complicated and taking fourth order derivatives and simplifying a very long differential equation is going to be tedious. Also, I need to do the above simplification for several examples, hence a mathematica code will be highly useful. $\endgroup$ – ramanujan_dirac Oct 2 '19 at 19:32
  • $\begingroup$ We cannot guess from your words what the equation looks like. $\endgroup$ – Alex Trounev Oct 2 '19 at 22:23
  • $\begingroup$ I don't think that really matters, but I have clarified my question by giving an example. Hope that helps. $\endgroup$ – ramanujan_dirac Oct 2 '19 at 22:47
  • $\begingroup$ diffEq /. f->(g[#] h[#]&)? $\endgroup$ – AccidentalFourierTransform Oct 3 '19 at 0:00

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