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I have a function $P(Q,x)$ ($P$ as a function of variables $Q$ and $x$) and I would instead like to known the function $Q(P,x)$ ($Q$ as a function of variables $P$ and $x$).

These functions are in general non-linear and quite complicated, but all variables and functions are real-valued.

Are there some standard procedures to solve these kinds of problems?

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    $\begingroup$ The inverse function theorem. Note that many proofs are actually constructive, in that it's possible to computationally approximate the inverse functions at particular points in practice. $\endgroup$
    – davidlowryduda ♦
    Commented Feb 2, 2022 at 21:43
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    $\begingroup$ You can think of the $x$ as a constant, so this is the same as finding an inverse of a single-variable function. You have an equation of the form $p=...$ that involves $q$, and you need to solve it for $q$ by rearranging it into the form $q=...$ with no $q$s on the right-hand side. $\endgroup$
    – Karl
    Commented Feb 2, 2022 at 21:43
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    $\begingroup$ No. There is no "standard" method. Almost all multivariate functions aren't globally invertible, anyway. $\endgroup$
    – K.defaoite
    Commented Feb 2, 2022 at 22:12

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