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

[annoymage]

Homework Statement

(2y-6x)dx + (3x+4x²y^-1)dy = 0

Homework Equations

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

 

[annoymage]

or am i suppose to use special integrating factor??

 

[Dick]

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.

 

[annoymage]

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]

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.