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I should use partial derivatives to find the differential equation that has the following solution:

$ u=f(x+\frac{y}{z},y+\frac{x}{z})$.

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There are infinitely many possibilities, naturally. The simplest one is a linear ansatz, $$ (\alpha\partial_x+\beta\partial_y+\gamma\partial_z) u=0 $$ i.e.,

α D[f[x + y/z, y + x/z], x] + β D[f[x + y/z, y + x/z], y] + γ D[f[x + y/z, y + x/z], z] // Simplify

which yields the equations

{z α + z^2 β - x γ == 0, z^2 α + z β - y γ == 0}

with solution $$ \beta \to -\frac{\alpha (y-x z)}{y z-x},\quad\gamma \to -\frac{\alpha (z^3-z)}{x-y z} $$ Of course, one parameter is arbitrary since the equation is linear. We can take $\alpha=x-yz$ to make the equation look as simple as possible,

(x - y z) D[f[x + y/z, y + x/z], x] + (y - x z) D[f[x + y/z, y + x/z], y] + (z - z^3) D[f[x + y/z, y + x/z], z] // Simplify
(* 0 *)

Final answer is $$ (x-y z)\partial_xu+ (y-x z)\partial_yu+z(1-z^2)\partial_zu=0 $$

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  • $\begingroup$ And what's the point of answering such irrelevant question, which is not even about Mathematica? $\endgroup$
    – yarchik
    Apr 28, 2022 at 11:52
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    $\begingroup$ @yarchik I interpreted the question as: how to solve this using Mathematica. Whether this was actually OP's intention or not does not seem all that relevant to me. $\endgroup$ Apr 28, 2022 at 11:54
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    $\begingroup$ What is relevant, however, is that the question should show at least a little bit of work. $\endgroup$
    – yarchik
    Apr 28, 2022 at 13:02
  • $\begingroup$ Unfortunately, this is not a PDE at all, since the arguments of the unknown function must be variables, and Mathematica says it in "The length of the derivative operator Derivative[0,1] in \!(*SuperscriptBox[\"f\", TagBox[RowBox[{\"(\", RowBox[{\"0\", \",\", \"1\"}], \")\"}],Derivative],MultilineFunction->None][x + *FractionBox[(y), (z)], y + *FractionBox[(x), (z)]]) is not the same as the number of arguments." when DSolve[(x - y z) D[f[x + y/z, y + x/z], x] + (y - x z) D[ f[x + y/z, y + x/z], y] + (z - z^3) D[f[x + y/z, y + x/z], z] == 0, f, {x, y, z}] is executed. $\endgroup$
    – user64494
    Apr 28, 2022 at 13:39
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    $\begingroup$ You are probably right with (x - y z) D[u[x, y, z], x] + (y - x z) D[u[x, y, z], y] + (z - z^3) D[u[x, y, z], z] == 0. $\endgroup$
    – user64494
    Apr 28, 2022 at 13:56

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