I am Learning Differential equation with ordinary differential equation. How to tell students the actual geometric meaning of differential equation? What is first order differential equation actually mean? what is nth order differential equation actually mean? I am confused how to tell these things. Please give some geometric meaning of these terms and also if possible some examples so that i can get an idea of these . In ordinary books just definition is given all of these but how to tell what geometrically they represent? Please help me. Thanks in advance.
You may have a look at Vladimir Arnold's book Ordinary Differential Equations, which geometric understanding is emphasised in.
I found this, a video lecture by MIT Open Course ware very helpful in understanding the geometric significance of $y' = f(x,y)$. It covers Direction fields and integral curves, enjoy!
Differential equations can have many different physical meanings. For example, dy/dx=f(x,y) graphed as a slanted line at every point (x,y) indicating the associated slope could indicate a current in water. If you have two metallic half-planes meeting at an angle, that same kind of graph can show you the rough shape of the electric field or electric potential(solution by conjugate functions).
Given dy/dx=f(x,y), you can consider a different differential equation, dy/dx=-1/f(x,y). The latter will be perpendicular to the former at every point.
As it happens, the point of closest approach on an ellipse to some point in its interior lies on a hyperbola passing through the origin horizontally, passing through the target point, and meeting the ellipse at the closest point at a right angle. So one method to find that closest point could be taking the ellipse's equation, expressing it as a differential equation, transforming it as above, then solving the resulting equation with initial conditions mentioned.
Asking "what does a differential equation really mean?" is the wrong question, if by "really" you mean "give me an exact physics equivalence". Things in math describe multiple things in physics, they "mean" multiple things at once -- this is the point of math, and the idea is that you can just do one mathematical theory to describe a bunch of different things.
Sometimes, the "multiple things at once" are explored within the math itself -- an example is the real numbers. What are the real numbers? Well, they're a set -- that's an unstructured, useless idea in itself. What does it "really mean"? Well, that question is answered by the variety of algebraic objects we can define out of this set -- you can define an additive group with the real numbers, then the real numbers are one-dimensional translations, you can define a multiplicative group, then they're scalings, you can define a ring or a field or one of those other things, and then they're in some sense objects on their own accord. And then there are a bunch of functions that map to the reals, so they also function as all sorts of things, like measures on sets and distances on metric spaces.
This kind of thing is I think where people get the idea that everything in math needs to have an exact physical equivalence, but there aren't really analogous structures defined for differential equations.
If you really want an answer to your question, the best I can give is "in general, differential equations are just recursive relations that describe the behaviour of continuous objects" -- it's the closest thing you can get to "induction on a continuum" or "recursion on the continuum", and they're just analogous to difference equations/recurrence relations on discrete sets. So if you know an initial state and you know the differential behaviour of an object -- as you often do in physics -- you're going to be using a differential equation. But they don't get more specific than that.