# Simplifying a differential equation acting on a product of functions

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 $$D'h=0$$?

• What is the question here? – Alex Trounev Oct 2 '19 at 19:19
• 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. – ramanujan_dirac Oct 2 '19 at 19:32
• We cannot guess from your words what the equation looks like. – Alex Trounev Oct 2 '19 at 22:23
• I don't think that really matters, but I have clarified my question by giving an example. Hope that helps. – ramanujan_dirac Oct 2 '19 at 22:47
• diffEq /. f->(g[#] h[#]&)? – AccidentalFourierTransform Oct 3 '19 at 0:00