Exact differential equation (introduction & example)

We will see the strategy of solving an exact differential equation from the idea of the total differential (aka total derivative) of a multi-variable function from calculus 3. An exact differential equation usually has the form of M(x,y)dx+N(x,y)dy=0. And we need to check if the mixed partial derivative of M(x,y) and N(x,y) is equal in order to use this approach of solving an exact differential equation.
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They're called exact equations because the total differential is also called the exact differential. It's exact because everything adds to a constant. The actual idea behind exact DE's is that there are two independent variables and no dependent variables, hence F being constant. This means that dF is always zero, but dx and dy are always changing but add up to zero exactly. When bprp did the calc one differentiation, he should have implicitly differentiated with respect to t, as if t was the independent variable, and then asked, "What's this t? I don't see any t's. I know that x and y are supposed to be functions of t, BUUUT, what if there was no t?" Then you multiply by dt to get rid of the dt's and, bam, an exact equation.

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In the derivative dy/dx it should be noted that x and y can have only values that are on the actual curve. How is this handled? If (x,y) coordinate value that is not on the curve is substituted to the dy/dx equation, then is that actually a derivatives that is resulted? Then derivative exists also outside the curves line?

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