# Intergrating factor

## Homework Statement

(2y-6x)dx + (3x+4x2y-1)dy = 0

## The Attempt at a Solution

(3xy+4x2)dy/dx = 6xy-2y2

i'm stuck, want to find the intergrating factor, it's xy2 when i reverse from the answer :D.. give me any hint please

## Answers and Replies

or am i suppose to use special integrating factor??

Dick
Science Advisor
Homework Helper
Are you sure there is an integrating factor? Maybe I'm just not seeing it. But you could also try the substitution v=y/x. That should make it separable.

yey, i got it now, there's another technique which i don't really know the details..

But to find suitable intergrating factor,

its something like

e($$1/M$$)[($$N(x,y)/dx$$) - ($$M(x,y)/dy$$)]

and is equal to xy2

and multiply both sides by the integrating factor, and will get Exact form of equation and solve it, hoho,

but maybe i need to try substituting v=y/x

HallsofIvy
Science Advisor
Homework Helper
Every first order differential equation has an integrating factor- it just may be hard to find!

Are you required to find an integrating factor? this equation is "homogeneous". Write it as
$$\frac{dy}{dx}= \frac{6x- 2uy}{3x+ 4x^2y^{-1}}= \frac{6- 2\frac{y}{x}}{3+ 4\frac{x}{y}}$$

Let u= y/x so that y= xu and dy/dx= u+ x du/dx and that will become a separable equation for u as a function of x. If you really want to find an integrating factor, I think that solving for u and then y and deriving the integrating factor from the solution is simplest.