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### Course: Differential equations>Unit 1

Lesson 7: Exact equations and integrating factors

# Exact equations intuition 2 (proofy)

More intuitive building blocks for exact equations. Created by Sal Khan.

## Want to join the conversation?

• Why are these differential equations called exact equations?
• At it is explained. Simply think of it as the difference between an exact equation and a non-exact equation. To be exact, the partial derivatives are equal to one another. p(x,y) = g(x,y) or M(x,y) = N(x,y). Hope that helps.
• at , why is ( My = Nx ) strong enough to imply an exact equation. It seems possible to me that My might equal Nx butthat no function Psi (P) exists such that M=Px and N=Py. (not withstanding that Fxy=Fyx for well behaved functons) Thanks
• This is under the premise that M is in fact Px in the first place. So if M=Px, then My=Pxy, and similar logic applies to N, which is Py (again may or may not be in the real world). So if My=Pxy and Nx=Pyx, and Pxy=Pyx, then My=Nx.
• I have three questions about PSI.

1. Is the symbol that we are using here for psi an upper case PSI (Ψ) or a lower case psi (ψ)?

2. I am also confused about the pronunciation of the letter. My teachers have always pronounced the letter psi as a hard s as in the word psychology. Sal has been pronouncing the letter as what sounds to me like the letter z in English, or like what I understand to be the sound of the Greek letter XI (Ξ) or xi (ξ). Are both pronunciations correct? Is one preferable to the other?

3. Is the letter psi used for some particular entity in mathematics, or perhaps in physics (or another science)?

Sal, thanks for all you are doing. I have just finished all of your calculus playlists, and I am looking forward to enjoying and learning differential equations from you.
• It isn't important that it's psi here, that's just common denotation. It's generally pronounced either with a hard s like psychology or as zi with a long i. The Greek letter xi is generally pronounced zee but is sometimes pronounced zi. It can get confusing which is why most people default to the hard s for psi.
• I understand the mathematical forms of exact equations, how to recognize them and how to solve them, but I'm still not sure when these equations are used? Can they be visualized or explained by a more solid example?
• what if the domain in the given function is discontinuous? then, the psi xy is not equal to psi yx. how can we test if it is an exact equation?
• Are "exact equations" and "conservative vector fields" the same thing?
• Although solving an exact equation and finding the potential function for a conservative vector field are very similar processes, exact equations and conservative vector fields are not exactly the same thing.

A vector field associates every point with a vector, whereas a differential equation is a relation between a function and its derivative(s). Good question, though!
• At how we obtained psi (x,y) = C? did he integrate?
• The solution you are looking for is always C. The derivative of a constant is zero, so when he has the derivative of psi = 0, this means Psi = C. So, yes, he did integrate.
• Are the partial derivatives commutative? e.g. (del(psi)/del(x)) of (del(psy)/del(y)) does that equal (del(psi)/del(y)) of (del(psy)/del(x))?
If not, then how can we represent it as multiplication?
• Great question! This can be answered by Schwarz's Theorem.

Schwarz's Theorem. If the second partial derivatives of your function are continuous at the relevant point, then fxy = fyx (order doesn't matter).

So I don't think we would call the partial derivatives commutative, but sometimes they do work that way.

Hope this helps,
- Convenient Colleague