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### Course: Differential equations>Unit 1

Lesson 7: Exact equations and integrating factors

# Exact equations example 2

Some more exact equation examples. Created by Sal Khan.

## Want to join the conversation?

• Is there a way to turn an equation, which is not exact, into an exact differential equation?
• Yes. That will be the subject of the next batch of videos when Sal starts talking about integrating factors.
• What year in college does an engineering major study differential equations? I am currently in ninth grade.
• Usually their 2nd or 3rd year of college. If you're taking AP calculus and do well on the AP exams you might take it your first year. Most engineering majors take Diff. Eqns. after they take 2 or 3 semesters of Calculus.
• Do all separable differential equations have My=Nx=0 ?
• Yes. The separable equation is in the form of
A(x) dx = B(y) dy
let's try to make it look like exact. (subtract B(y) dy from both sides)
A(x) dx - B(y) dy = 0
let's now check if My = Nx, or dA(x)/dy = -dB(y) / dx
derivative of a function of x with respect to y is equal to zero, or vice versa. So,
My = 0, Nx = 0 thus My = Nx so yes, all separable differential equations have My=Nx,
or in other words separable differential equations are a subset of exact equations, or every separable equation is also an exact differential equation but not every exact differential equation is necessarily separable differential equation.
• In the first example of this video (when he started solving at about ), the differential equation is a separable differential equation. So, how is the name of this equation, should I say 'separable' or 'exact'?

Thank you!!
• You say, that this is an exact equation, because My=Nx, and separable, because you can algebraically sweep x and y to different sides of the equation. Two different definitions of two properties, and both match.
• Why is this kind of differential equation called "exact?" Is there some significance to that term that might apply elsewhere, or is it just a term that was coined to describe this particular situation?
• The equation is called Exact since it is the exact differential of a function. This function (Psi) can be obtained by the process taught in the videos.
(1 vote)
• Are people confused by the fact that the TOTAL derivative notation looks like the PARTIAL derivative notation?
• With respect to the topic "Exact Equations Example 2" I tried a different approach. Instead of choosing M(x,y) I started with N(x,y). So:
ᶴΨydy = ᶴN(x,y)dy = Ψ(x,y)
Then I differentiated the result with respect to x. The new expression should be equal to M(x,y), so is possible to define d/dxf(x) and after f(x). Finally I got the same result Mr. Sal had achieved in his video class.

Ney
• Around , Sal's second question (2x + 4y) + (2x - 2y)y' = 0.
So it's not an exact equation. Can you solve by separation?
Mainly I don't know what to do with (2x - 2y)y'.