 # 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 $M\mathrm{d}x+N\mathrm{d}y=0$
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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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
${y}^{\prime }=y\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}{y}^{\prime }-y=0\stackrel{\cdot {e}^{-x}}{\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}}\frac{{y}^{\prime }}{{e}^{x}}-\frac{y}{{e}^{x}}=0$

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