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Exact Differential Equations and Integrating Factors 

classification: this technique applies to firstorder differential equations 
\( M(x,y)~dx + N(x,y)~dy = 0 \) and \(M_y = N_x\) 
Exact differential equations are firstorder differential equations of the form \(\displaystyle{ M(x,y)~dx + N(x,y)~dy = 0 }\) where \(M_y = N_x\). The requirement that \(M_y = N_x\) means the differential equation is what we call exact. Here are some equivalent ways of writing this differential equation.
\(\displaystyle{ M(x,y)~dx + N(x,y)~dy = 0 }\) 
\(\displaystyle{ M(x,y) + N(x,y)y' = 0 }\) 
\(\displaystyle{ M(x,y) + N(x,y)\frac{dy}{dx} = 0 }\) 
How To Solve 
Lets jump right into a video to see how we solve these. Here is a great video clip explaining the technique and showing a couple of examples.
Dr Chris Tisdell  exact equations  
In the video above, he does a good job of emphasizing that, before you start actually solving the differential equation, you need to make sure it is exact by calculating \(M_y\) and \(N_x\) and checking that they are equal. Otherwise, this technique will take you down a path that leads nowhere. Most instructors will take off points if you don't show work that you did this. So check with your instructor. But even if they don't take off points, it could save you a lot of time on homework and exams if you do this and it takes only a few seconds. So it is a good investment of your time.
The main difference with these types of problems is that you need to remember the constant of integration is actually a function of integration. This occurs because we are taking the partial integral of one variable while holding the other variable constant. This concept is covered in detail on the multivariable partial integration page.
Here is another video that is very similar to the first one, with a few different examples. But watching this one also, will help you understand this technique better.
Dr Chris Tisdell  more exact equations  
Integrating Factors 
If we have a firstorder equation of the form \( M(x,y)~dx + N(x,y)~dy = 0 \) but \(M_y \neq N_x\), i.e. the equation is not exact (also called inexact), we may be able to convert the equation to exact using integrating factors. You first came across integrating factors when you studied linear, firstorder equations. This technique also uses an integrating factor but it is calculated differently.
There are actually two possible integrating factors that convert the differential equation to an exact equation.
\(\displaystyle{ \mu_1 = \exp \int{ \frac{M_y  N_x}{N} dx } }\) 

\(\displaystyle{ \mu_2 = \exp \int{ \frac{M_y  N_x}{M} dy } }\) 
The trick comes in when you are asked to evaluate these integrals. The evaluation is not always possible and can get quite messy. Also, one integral may yield an integrating factor that is quite complicated while the other one may be much easier to use. Regardless, these integrating factors can, at times, be quite useful. Let's look at an example in this next video.
Khan Academy  Integrating Factors  
Okay, time for some practice problems.
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Practice Problems 

Instructions   Unless otherwise instructed, solve these differential equations. Make sure to check that the equation is exact before attempting to solve. Give your answers in exact form.
Level A  Basic 
Practice A01  

\(2xy~dx+(x^2+3y^2)~dy=0\)  
solution 
Practice A02  

\(2xy~dx+x^2~dy=0\)  
solution 
Practice A07  

\( (2xy+1) + (x^2+3y^2)y'=0 \)  
solution 
Practice A08  

Find the explicit solution for the exact differential equation \( (e^{x+y}2x)dx + (e^{x+y}+2y)dy = 0\), \(y(0)=0\).  
answer 
solution 
Level B  Intermediate 
Practice B01  

\((3x+y+1)~dx+(3y+x+1)~dy=0\)  
solution 
Practice B02  

\((2x+y+1)~dx+(2y+x+1)~dy=0\)  
solution 
Practice B03  

\((3x^22xy+2)~dx+(6y^2x^2+3)~dy=0\)  
solution 
Practice B04  

\( (2xy^{1}+2ye^{2x}1) +\) \( (e^{2x}+y^2x^2y^{2})\frac{dy}{dx}=0 \)  
solution 