How to find all $f:{\Bbb R^3} \rightarrow {\Bbb R}$ of class ${\mathcal C}^1$,

$(z-y)\frac{\partial f}{\partial x} + (x-z)\frac{\partial f}{\partial y} + (y-x)\frac{\partial f}{\partial z}=0$ ?

I don't how to start. Is there any method to solve this kind of PDE ?


Note that this of Lagrange’s form: a linear PDE of the first order. Note that an equation of the form $Pp+Qq=R$ where $p=u_x, q=u_y$ is given by solving the auxiliary equation $$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$$

In this case, taking $f$ as a function of the independent variables $x,y,z$, we have, the auxiliary equation as: $$\frac{dx}{z-y}=\frac{dy}{x-z}=\frac{dz}{y-x}=\frac{df}{0}$$ giving us $df=0, dx+dy+dz=0, xdx+ydy+zdz=0$. Integrating, we get, $$f=c_1, x+y+z=c_2, x^2+y^2+z^2=c_3$$

Hence, the solution is $$f=F(x+y+z,x^2+y^2+z^2)$$ where $F$ is an arbitrary function.


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