The first alternate notation I think in maths, I used was dot (period) notation for scalar multiplication. Next would be different versions of standard form (aka engineering notation).
Later, at high-school (UK, 16-18yo) we used both d/dx type notation for differentiation and x-dot (a dot above the letter for each level of differentiation).
There would be 3 versions of x= ut + (at^2)/2 that we'd know, this one, d/dx, and dot-notation; primarily the former was used in physics and the latter in maths. We'd also use suffixed numbering, u_0, u_1 on occasions and use u, v, w on other occasions.
One important point - they're not different equations, just written differently.
You write for your audience, in an advanced paper you can abbreviate and use shorthand because you know the reader (in general) will understand.
As an undergrad I found that each department I studied in would tend to use different notation - chemistry, ecology, maths, physics, computing - for the same things. Though sometimes different professors would buck the trend ... they also had different accents and vocabulary, our QFT professor loved the word recapitulation.
You also get different symbol use, like rho for density in one domain and d in another; but that's not really notation per se.
TL;DR sometimes you come across different notation earlier. Different notation doesn't change the equation.