# Thread: Finding the Integrating Factor

1. ## Finding the Integrating Factor

Question: For what values of $\displaystyle m$ and $\displaystyle n$ will $\displaystyle u = x^ny^m$ be an integrating factor for the differential equation

$\displaystyle (-3y-2x)dx + (5x+4x^2y^{-1})dy = 0$

The exact differential equation which results from multiplying by this integrating factor has solution $\displaystyle F(x,y) = C$ where $\displaystyle F(x,y) =$___________________________.

Attempt at solving the question:

First I check to see whether either differential is exact (equal to each other), of course they are not.

The continue to solve and reach this point:

$\displaystyle u'(y)(-3y -2x) + u(y)(-3-2x) = u(y)(5+8xy^{-1})$
$\displaystyle (-3y -2x)u'(y) + (-3-2x-5-8x^{-1}) = 0$
$\displaystyle (-3y -2x)u'(y) + (-8-2x-8xy^{-1}) = 0$

I should be able to factor something out, but i don't seem to find any, so am I doing the question wrong?

2. did you already solve this? i saw the "solved" tag above but i don't actually see that you solved your problem.

3. Actually, i did that because no one was responding. Anyways, I did figure out my integrating factor to be $\displaystyle x^{-1}, n = -1$

I get to a point where after i multiplied both parts of the equation by the integrating factor and then i took the integral of the part of the equation and i get my F(x,y) which is $\displaystyle -1/7x^4y^7+ (1/3)x^9y^8 + x^6/6 + K(y)$. After wards i took the partial derivative with respect to y and got $\displaystyle -x^4y^6 + (8/3)x^9y^7 + k'(y)$. I equate both equations and i get $\displaystyle K'(y) = 0$, which if i integrated k'(y), i would get C since. But my ans i entered was incorect.

$\displaystyle F(x,y) = (-1/7)x^4y^7 +(-1/3)x^9y^8 (-1/6)x^6 = C$

4. you can actually find the integrating factor of the form $\displaystyle \mu(x,y)=x^my^n,$ the equation has it when for $\displaystyle m,n\ne0$ constants we have $\displaystyle \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}=m\dfrac{N}{x}-n\dfrac{M}{y}.$

it's a necessary and sufficient condition, so if it doesn't hold, don't waste your time.

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on the other hand always compute $\displaystyle \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}$ and see its form, it would give ya an idea if you can find an integrating factor depending exclusively of $\displaystyle x$ or $\displaystyle y.$

here's a hint, multiply both sides by $\displaystyle y$ and apply the method to find the integrating factor you were asked to find.