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### Course: Differential equations>Unit 1

Lesson 7: Exact equations and integrating factors

# Exact equations intuition 1 (proofy)

Chain rule using partial derivatives (not a proof; more intuition). Created by Sal Khan.

## Want to join the conversation?

• Which Calc videos should I watch if I struggled a bit with this video?
• Partial Derivatives 1 and 2.
• I'm not quite understanding where the function Psi came from. This is probably a stupid question but why are we introducing Psi, a 3 dimensional function and 3 dimensional calculus along with it, into 2 dimensional differential equation problems?
• He's just using it as an example. He really could have picked any Greek letter and defined it as a function... he chose Psi.
• Seeing y as a function of x seems to eliminate y's independence. Doesn't Psi then become effectively a function of just one variable? If x and y were truly independent variables, it seems that all of this math would change drastically. (ie partial deriv of psi wrt y would no longer be multiplied by dy/dx since dy/dx would be undefined.)
• Sal addresses this right at the end... If y is not a function of x then dy/dx will equal zero and then second term would then be zero and dΨ/dx would just equal the partial derivative of Ψ with respect to x.
• -
please explain why he says dy is differnt from ∂y..
thanks, pl rply asap
• So dy is from regular derivatives and ∂y is from partial derivatives. I first encountered them in multivariable calculus. It is what you need to use if you have multiple variables. Psi is the overall function. It is taking 2 pieces: x, y. Y is also a function of x. Since Psi takes 2 inputs it makes sense that you would need to differentiate each of them and the tool we have for that is partial differentiation. Additionally since y is a function of x you need to derive that and since it is only one variable you have dy/dx attached to that partial derivative.

Note: I am only learning this now myself but this is why it seemed to make sense to me. Hopefully someone will give a more authoritative answer eventually.
• Can you write all functions in the form H(x,y) = f1(x)g1(y) + f2(x)g2(y) + ... ?
• I don't think that this is possible. How about H(x,y)=sin(x*y)? I don't think that this can be represented as a sum of products of functions of x and y.
• Can any one recommend me a good book for the differential equations ? Thx in advance :D
• Elementary Differential Equations with Boundary Value Problems by Boyce and Diprima is the best one I have used
• Is it just my computer or is there a problem with the video quality?
• Yes some of the old videos of Khan Academy has bad quality, unfortunately.
• Why exactly did Sal use the Greek letter Psi? Couldn't he have used something else like alpha, beta, epsilon, eta?
• He could have yes. Psi is just a variable and it doesn't matter what you call it. What you use is up to personal preference so I would assume Sal like Psi. You also want to avoid reusing variables since then new material can get confusing.
• at , why are there n terms in the f(x)g(x). Its an extremely confusing example