Integrating Factor and Potential Function for Non-Exact Multi-Variable ODEs

In summary: Well, multiplying by xy is a no-brainer to get rid of the fractions. Sometimes if an equation is not exact, you can find an integrating factor that is a pure function of x or y. If you are interested, I printed an article about that that you can read as post #2 in the thread:https://www.physicsforums.com/showthread.php?t=342132As to how to solve the equation once you have it exact, look at example 2 inhttp://tutorial.math.lamar.edu/Classes/DE/Exact.aspxfor an easy to read example.That example was incredibly helpful. I was overcompl
  • #1
sleventh
64
0
Hello All,
Given the equation (2/y + y/x)dx + (3y/x + 2)dy
I am first asked to show the equation is not exact. To do this I showed the mixed partials were not equal i.e.:
(2/y + y/x)dy != (3y/x + 2)dx

I am then asked to find an integrating factor and show the potential function is given by
f = x^2 + y^3 + xy^2

I have consulted my ODE's textbook but can not see how to apply the methods there to a multi-variable problem. I also haven't had much luck online.

Thank you for your help
 
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  • #2
sleventh said:
Hello All,
Given the equation (2/y + y/x)dx + (3y/x + 2)dy

That isn't an equation; it is just an expression. Perhaps you want to set it = 0.
I am first asked to show the equation is not exact. To do this I showed the mixed partials were not equal i.e.:
(2/y + y/x)dy != (3y/x + 2)dx

I am then asked to find an integrating factor and show the potential function is given by
f = x^2 + y^3 + xy^2

I have consulted my ODE's textbook but can not see how to apply the methods there to a multi-variable problem. I also haven't had much luck online.

Thank you for your help

Try multiplying your equation by xy to clear the fractions. Also, since you are given a potential function, you could work backwards to see what to do.
 
  • #3
thank for your reply LCKurts
I see how multiplying by xy allows for an exact equation (and i think it's assumed to equal zero) since the mixed partials are equal. But to be honest, I do not see how you solve for the integrating factor other then guess by experience nor do I see how to actually solve for f(x,y) once you have your integrating factor. Thank you for any help.
 
  • #4
sleventh said:
thank for your reply LCKurts
I see how multiplying by xy allows for an exact equation (and i think it's assumed to equal zero) since the mixed partials are equal. But to be honest, I do not see how you solve for the integrating factor other then guess by experience nor do I see how to actually solve for f(x,y) once you have your integrating factor. Thank you for any help.

Well, multiplying by xy is a no-brainer to get rid of the fractions. Sometimes if an equation is not exact, you can find an integrating factor that is a pure function of x or y. If you are interested, I printed an article about that that you can read as post #2 in the thread:

https://www.physicsforums.com/showthread.php?t=342132

As to how to solve the equation once you have it exact, look at example 2 in

http://tutorial.math.lamar.edu/Classes/DE/Exact.aspx

for an easy to read example.
 
  • #5
That example was incredibly helpful. I was overcomplicating but now see it is in the fact
eq0012MP.gif


can be written from
eq0011MP.gif


looking at the single variable case i thought you had to use the e^int(p(x)) expression, but that is just a simple case used as example.

One last question i do not see how multiplying by a integrating factor allows for a general solution. Wouldn't you be changing your function by doing this?
 

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What is an Integrating Factor?

An integrating factor is a mathematical tool used in solving differential equations. It is a function that is multiplied to both sides of a differential equation to make it easier to solve.

Why is an Integrating Factor used?

An integrating factor is used to transform a non-exact differential equation into an exact one. This makes it easier to solve the equation and find a general solution.

How is an Integrating Factor found?

An integrating factor can be found by using different methods such as the method of variation of parameters, the method of undetermined coefficients, or by using specific formulas for different types of differential equations.

What are the benefits of using an Integrating Factor?

The use of an integrating factor can simplify the process of solving differential equations, making it more efficient and accurate. It can also help in finding a general solution to the equation, which can be used to solve specific problems.

Are there any limitations to using an Integrating Factor?

Yes, there are some limitations to using an integrating factor. It may not always be possible to find an integrating factor for a given differential equation, and even if it is possible, the process may be complex and time-consuming. Additionally, the use of an integrating factor may not always result in a unique solution.

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