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

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In summary, the conversation discusses how to solve the differential equation (xy2+y2)dx + xdy = 0. The first question asks if the equation can be an exact differential equation, to which the response is no. The second question asks for help in finding an integration factor to change the equation into an exact one, but the link provided does not work. The experts suggest using a different method to solve the equation and await the lecturer's solution method.
  • #1
adrianwirawan
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(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..

:cry:
 
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  • #2


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?
 
  • #3


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?
 
  • #6


Hmmm, I was allso sleepy :blushing: link doesn't work...
 
  • #7


micromass said:
Hmmm, I was allso sleepy :blushing: link doesn't work...

The link seems to work for me. It's just that the method doesn't work for this problem.
 
  • #8


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:

[tex]-\frac{x+1}{xy}dx + \frac{x + \ln(x) -1}{y^2}dy = 0[/tex]

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.
 

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

What is an ODE?

An ODE (Ordinary Differential Equation) is a mathematical equation that relates a function to its derivatives. It is commonly used to model physical phenomena in various scientific fields.

What is an exact solution?

An exact solution to an ODE is a solution that satisfies the equation without any approximation or error. It is the most accurate representation of the relationship between the function and its derivatives.

3. How do you solve an ODE?

To solve an ODE, you can use various analytical or numerical methods. One common method is the separation of variables, where you separate the variables on each side of the equation and integrate both sides to find the solution.

4. What is the general form of an exact solution for the ODE (xy^2 + y^2)dx + xdy = 0?

The general form of an exact solution for this ODE is given by y = (C - x^2) / (x^2 + 1), where C is a constant.

5. How do you check if a solution is exact for the ODE (xy^2 + y^2)dx + xdy = 0?

To check if a solution is exact, you can use the method of verifying the exact equation. This involves taking the partial derivatives of the solution with respect to x and y and then substituting them into the original ODE. If the resulting equation is true, then the solution is exact.

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