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### Course: Differential equations>Unit 1

Lesson 7: Exact equations and integrating factors

# Integrating factors 2

Now that we've made the equation exact, let's solve it! Created by Sal Khan.

## Want to join the conversation?

• This was completely new to me. What I expected to see was a final step "undoing" the factor of integration. The original differential equation was altered so it was a different problem, wasn't it? How come the solution to this different problem is also a solution to the original problem?
• Consider the equation y=3x. If you multiply both sides by 5, you have 5y=15x. Now plug in 2 for x into both equations, what is your answer? You get y=6 in both equations since from algebra we can see that multiplying both sides of the equation by a number doesn't change it! =D
• In , mu is used. Why mu? Isn't mu used for an arithmetic mean in statistics?
• There aren't enough letters in either the Latin or Greek alphabets for mathematicians, so when there is no fear of being misunderstood we'll use letters that are "already used". If you think it's weird now, just wait until you take a course in abstract algebra where pi is often used as a variable for an arbitrary permutation. :)

Honestly, I don't know why mu is chosen, but it's the way they go in every ODE book I've ever read. I suppose it was the choice the original mathematician used in the original paper on the subject ("mu for multiplication factor"?) and it's as good a letter as any so no one has changed it.
• Is it possible for there not to be an integrating factor?
• Yes, in that case it is either an integrating factor of 1, so you can leave it out, or if no integrating factor exists, it is not solvable through the exact equation method.
• i m nt able to solve (x^3y^3+)dx + x^4y^2dy = 0 using this method
• This equation is separable so you don't need this method
• Can EVERY non-exact differential equation be transformed to an exact one, by multiplying it with some integration factor?
• According to the video, does Sal means there is no absolute integrating factor? For example it could be a function of x or y.
• There can be many integrating factors that cause the equation to be exact, but some are easier to find than others
(1 vote)
• AT what happened to the y' how come it did not come down into the new equations, specifically to the right of the equals
• Because in the method to solving Exact Equations, you consider what is multiplying your `y'` as your function `N(x,y)` and the rest as your function `M(x,y)`. The original `y'` is used just to separate those 2 parts, it no longer has any involvement in the rest of the solution.