For Leibniz notation, we know what y is differentiated in terms of, e.g. dy/dx. However, for Lagrangian notation it's just y'. How will we know what y is differentiated in terms of for a differential equation like y' + xy = 0?

For Leibniz notation, we know what y is differentiated in terms of, e.g. dy/dx. However, for Lagrangian notation it's just y'. How will we know what y is differentiated in terms of for a differential equation like y' + xy = 0?
You learn to read problems, equations and diagrams. You learn it, become familiar with it and it becomes second nature. And sometimes often you scratch your head and say what the %$&?%? is this? And you grunt it out. And sometimes there are two, three or ten of you looking at the same thing trying to interpret the thing. (by then it is usually more than a y' thing.
Generally. It's pretty obvious what it means. I cannot recall a situation where I was left in any doubt. Usually thought the dot notation seems to be differentiation wrt to time, but the context makes it obvious in most situations.
If in doubt go with differentiation wrt time. Generally though it is pretty obvious in the context. Do you a particular example that is bothering you?
It's implied. Note that y' can't be used for a partial derivative. That means that it must be a total derivative with respect to the independent variable. In this case that's x.
If there are more than two variables then you're not allowed to use the prime notation by convention. You then have to switch to partial derivatives because you've then gone into multivariable analysis in which the differential equations are partial differential equations.
Ah thanks, I guess that should had been obvious. Thanks again to everyone!
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