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For differential equations, when trying to determine the uniqueness of an equation in the form dy/dx = q(y)p(x) (where p(x) and q(y) are any functions of x and y, respectively), is there any particular reason why dy/dx = f(x,y) = q(y)p(x) is then later differentiated with respect to y as opposed to x? Why take the partial differential equation in the form ∂f/∂y instead of ∂f/dx? Is the only reason to determine where the range is continuous?

What's the intuitive and mathematical reasoning behind ∂f/∂y determining uniqueness as opposed to integrating dy/dx and then looking for values of y that lead to discontinuities?

What does ∂f/∂y represent by itself? Is it just how the slope changes wrt y or is there any other implication?

Also, is there a general trend or rule to determine existence and uniqueness for a whole family of DEs (i.e. for nth order, linear or nonlinear DEs)?

Any help and explanation would be greatly appreciated! Thank you!

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# Explanation for uniqueness

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