Why Is ∂f/∂y Used to Determine Uniqueness in Differential Equations?

[MathewsMD]

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!

 

[gtoguy97]

The reason for taking the partial derivative of f(x,y) with respect to y is to determine how the slope of the line changes with respect to y. This can help to determine if the range of an equation is continuous or not. Taking the partial derivative of f(x,y) with respect to y gives us an expression for the instantaneous rate of change of y, which is often referred to as the "slope" of the line. This expression can then be used to determine if the range of the equation is continuous or not. If the range of the equation is discontinuous, then the equation does not have a unique solution.The intuitive and mathematical reasoning behind ∂f/∂y determining uniqueness is that it allows us to determine how the slope of the line changes with respect to y. If the slope changes with respect to y, then it is possible that the equation may not have a unique solution.The partial derivative of f(x,y) with respect to y gives us an expression for the instantaneous rate of change of y, which is often referred to as the "slope" of the line. This expression can then be used to determine if the range of the equation is continuous or not. If the range of the equation is discontinuous, then the equation does not have a unique solution.In general, there is no single rule or trend that can be used to determine existence and uniqueness for a family of differential equations. Different types of equations may require different methods in order to determine existence and uniqueness.