# What can we say about the solution of this PDE?

by mousakas
Tags: solution
 P: 5 Hello! I would like to find some functions F(x,y) which satisfy the following equation $\frac{F(x,y)}{\partial x}=\frac{F(y,x)}{\partial y}$ For example this is obviously satisfied for the function $F= exp(x+y)$ I would like however to find the most general closed form solution. Do you have any ideas? Could it be that it has to be a function of x+y only for example? I tried to get some info by taylor expantions but I was not so succesful.
 Emeritus Sci Advisor PF Gold P: 4,500 What can we say about the solution of this PDE? Let u=x+y and v=x-y. Then F(x,y) can be re-written as F(u,v) and $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial u} + \frac{\partial F}{\partial v}$$ $$\frac{\partial F}{\partial y} = \frac{\partial F}{\partial u} - \frac{\partial F}{\partial v}$$ So for these to be equal we get that $$\frac{\partial F}{\partial u} + \frac{\partial F}{\partial v}= \frac{\partial F}{\partial u} - \frac{\partial F}{\partial v}$$ which reduces to $$\frac{\partial F}{\partial v}=0$$ so F is a function of u only (i.e. F can be written as F(x+y)) This is a fairly common technique for finding the solutions to differential equations like this - divine what the answer should be then use a change of variables to prove it
 P: 5 Thank you both for your answers :) BUT look also that the function in the r.h.s. is not F(x,y) but F(y,x) For example if $F(x,y)=\frac{x}{x+y}$ then $F(y,x)=\frac{y}{x+y}$ That's what confuses me.
 Emeritus Sci Advisor PF Gold P: 4,500 Ahhh, my mistake, I misread the question. Let's define $$G(x,y) = \frac{\partial F(x,y)}{\partial x}$$ Then all the equation in the OP is saying is that G(x,y)=G(y,x). So G is any function which is symmetric in x and y. Then integrating w.r.t x says that $$F(x,y) = \int G(x,y) dx + H(y)$$ integrate G with respect to the x variable. The "constant of integration" in this case is is a function which is constant in x, so can be any function of y. An example of a solution: Pick G(x,y) = x2+y2. Then $$F(x,y)=\frac{x^3}{3}+y^2x+H(y)$$ where H(y) is any function you want.