# Solving this First-Order ODE

• 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:

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

a) Show that the given equation is not exact.
b) Find constants a and b for which $\displaystyle t(x,y) = x^a*y^b$ 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
if there's constants $\displaystyle m,n\ne0$ that verify $\displaystyle \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}=m\dfrac{N}{x}-n\dfrac{M}{y},$ the integrating factor is $\displaystyle \mu(x,y)=x^my^n.$
Have you tried to re-arrange the original equation to the form $\displaystyle y'+f(x)y = g(x)$ ?