Solving this First-Order ODE

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• Sep 29th 2010, 10:40 AM
AKTilted
Solving this First-Order ODE
I've been having some real trouble solving this differential equation, mainly because I can't figure out what method to use. The question is as follows:

Consider the differential equation:

$(x^3/y + 3/x)dx + (3x^3 - x^4/(2y^2) - 1/(2y))dy = 0$

a) Show that the given equation is not exact.
b) Find constants a and b for which $t(x,y) = x^a*y^b$ is an integrating factor for the given equation.
c) Use part b) to solve the given equation.

I think it's easy to show that the ODE is not exact but I'm not sure how to do the rest...

Thanks a lot,
AK
• Sep 29th 2010, 01:01 PM
Krizalid
make your life easier:

if there's constants $m,n\ne0$ that verify $\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}=m\dfrac{N}{x}-n\dfrac{M}{y},$ the integrating factor is $\mu(x,y)=x^my^n.$
• Sep 29th 2010, 02:12 PM
AKTilted
Ok, so I used that equation and determined that m = -3 and n = 1/2 for the given ODE. How do I use the integrating factor u(x,y)=x^-3*y^(1/2) to solve the ODE. I'm a little bit confused...
• Sep 29th 2010, 02:44 PM
pickslides
Have you tried to re-arrange the original equation to the form $y'+f(x)y = g(x)$ ?
• Sep 29th 2010, 03:02 PM
AKTilted
yes, i've tried to reduce it to that form, but I can't successfully move the variables to match that kind of form of ODE.