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### Course: Differential equations>Unit 1

Lesson 7: Exact equations and integrating factors

# Exact equations example 3

One more exact equation example. Created by Sal Khan.

## Want to join the conversation?

• In the final part why does he differentiate the y values if he is implicitly differentiating with respect to x ?
• y is a function of x. If you notice, he is differentiating everything according to x, (the total derivative, not the partial). He is basically applying the chain rule.

`d/dx f(y) = f'(y)dy/dx` when y is a function of x.
• At , isn't he using the product rule rather then the stated chain rule?
• Well, he's using the product rule to differentiate x^2y as well as the chain rule to differentiate y which is a function of x.
• At , Sal divides the differential equation by dx. But, I thought that derivatives such as dy/dx are not supposed to represent fractions? Are we actually dividing dy by dx to get dy/dx or is there a more subtle meaning and Sal is using an abuse of notation?
• dy/dx is a fraction. dx is just an infinitesimal change in x, while dy is an infinitesimal change in y. dy/dx is just the same thing as Δy/Δx, as Δx approaches 0. There is no abuse of notation. This is actually what the derivative means.
• Can't you just separate the equation at ?
• Yes you can but here Khan is just trying to show how to use the Exact Equation approach.
• at why are we even using the product rule? when I put d/dx(x^2*y) into Wolfram Alpha it gives me 2xy as the answer. This is very confusing to me.
• Why is y a function of x at and not just another variable? Shouldn't the partial derivative of Psi with respect to x hold y as a constant?
• lets say we have a equation such as (2x+y) - (x+6y)y' = 0. Would this equation be exact? How does the minus in M(x,y) "-" N(x,y)y' = 0 affect the question?
(1 vote)
• you can distribute the "-" inside and get (2x+y) + (-x-6y)y' = 0 which if you take partial respect to y for M you will get 1 and partial respect to x for N is -1 so they are not equal to eachother therefore it is not an exact equation.
• Do I have to take My first and then Nx or can I also take Mx first and then Ny and look if it's equal?
(1 vote)
• Yes you can. The coefficient of dy/dx is always the Psi(y) term or N and the other is the Psi(x) or M term. As long as you take Nx and My, you will get Psi yx= Psi xy and can prove its exact. Hope this helps