This is rather standard, though, unfortunately this knowledge is confined to a rather narrow segment of mathematicians working on PDEs. The main reason is that, most of the time, this is not necessary for practical problems. In fact there are multiple ways converting a PDE into invariant form.
In mathematical physics, this question comes up when trying to generalized equations from flat space to curved space. The main technical device is to introduce explicit dependence on a metric $g$ and use covariant differentiation, with respect to $g$, to replace ordinary derivatives. The invariant form of the original flat space equations is then obtained by simply replacing $g$ by the Euclidean metric. This method is sometimes colloquially known as the comma goes to semicolon rule, where the punctuation signs represent notation for ordinary and covariant derivatives. More on this can be found in textbooks on GR. For instance, Gravitation by Misner, Thorne & Wheeler would definitely discuss this.
I would guess that you might be happy to stop here. But if you are looking for more generality and mathematical sophistication, read on.
Another method is to repeatedly introduce auxiliary dependent variables, until the PDE system becomes first order and this can be done in such a way that all dependent variables are actually differential forms and differentiations are only effected through the exterior derivative. Once in this form, the PDE system is manifestly invariant. This approach was pioneered by Cartan and is explained in depth in the well known book Exterior Differential Systems by Bryant et al.
Yet another way, which does not involve reduction to a first order system, is to notice that derivatives of arbitrary order are mathematically described in an invariant way by jets. An arbitrary PDE can then be described in terms of jets. Once one unwinds all the relevant definitions, this kind of description is manifestly invariant, but not necessarily always helpful, since all it really does is bundle up the complicated rules for transforming higher derivatives under coordinate changes into the language of jets. Nevertheless, I think it is important to learn about it, as it provides the tools to study PDE systems in a very deep way. An elementary treatment of PDEs in the context of jets can be found in the book Applications of Lie Groups to Differential Equations by Olver. A much more sophisticated treatment can also be found in the aforementioned book by Bryant et al. Another very helpful reference is Natural Operations in Differential Geometry by Kolar, Michor & Slovak.