Why aren't exact differential equations considered PDE? In the exact differential equation Mdx+Ndy=0

IJzerboor07 2022-09-11 Answered
Why aren't exact differential equations considered PDE?
Exact differential equations come from finding the total differential from some multivariable function.
In the exact differential equation M d x + N d y = 0
M and N are considered to be partial derivatives of some potential function... So why aren't exact differential equations considered PDEs? After all, you're finding the potential function given it's partial derivatives...
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Answers (1)

Monserrat Ellison
Answered 2022-09-12 Author has 22 answers
Step 1
Because the partial differentials part is just a method of solving them, it's in the intermediate steps of a solution, not in the DE itself from the start. A bad example(can't think of a better one right now) would be considering x 2 = 0 a second degree polynomial because you can introduce parameters and make it x 2 = 4 , x > 0.
Step 2
Also, consider being able to solve a D.E. by transforming it into exact equation by multiplying it with an integrating factor or by using another method that has nothing to do with partial derivatives. Why would you call that a PDE?
A more specialized example would be
y = y y y = 0 e x y e x y e x = 0

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