Solving ODE: (xy^2 + y^2)dx + xdy = 0 - Exact Solutions

[adrianwirawan]

(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..

 

[LCKurtz]

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?

 

[adrianwirawan]

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?

 

[micromass]

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...

 

[LCKurtz]

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.

 

[micromass]

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

 

[LCKurtz]

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]

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.