# Show that this ODE: (xy^2 + y^2)dx + xdy = 0, can be an exact ODE and its solutions!

(xy2+y2)dx + xdy = 0

the questions are:
a. Show that the equations above can be an exact differential equations!
b. Determine its solutions!

Help me please because i have working on it for 3 hours and i can't find its integration factor to change the un-exact differential equation above into an exact one. I need your help..

## Answers and Replies

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LCKurtz
Homework Helper
Gold Member

That equation isn't exact. But instead of trying to find an integrating factor, why don't you just separate the variables, which is easy?

Thanks LCKurtz, but my lecturer ask me to solve it by finding its integration factor.. I wonder how it is.. Can you give me a clue?

LCKurtz
Homework Helper
Gold Member

The following site gives you some information on how to determine an integration factor; http://www.sosmath.com/diffeq/first/intfactor/intfactor.html

Case 2 is of interest here...
I don't think so unless I was sleepy when I checked it and am still asleep. I'm guessing the problem has a typo.

Hmmm, I was allso sleepy link doesn't work...

LCKurtz
Homework Helper
Gold Member

Hmmm, I was allso sleepy link doesn't work...
The link seems to work for me. It's just that the method doesn't work for this problem.

LCKurtz
Homework Helper
Gold Member

For what it's worth, and probably not very relevant, here's an exact DE that has the same solution set f(x,y) = C:

$$-\frac{x+1}{xy}dx + \frac{x + \ln(x) -1}{y^2}dy = 0$$

I don't see any obvious way to manipulate the original DE into this form.

To the original poster: I hope you will report back what your lecturer gives for a solution method.