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Hi,

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!

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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