I need to find integrating factor, i will solve the equation once the integrating factor is found.

Equation:

$\displaystyle (10x^2+2xy+6y^2)dx+(3x^2+4xy+5y^2)dy=0$

Where the equation is in the form of:$\displaystyle (M)dx+(N)dy=0$

$\displaystyle \dfrac{d}{dy}[M]=2x+12y$

$\displaystyle \dfrac{d}{dx}[N]=6x+4y$

$\displaystyle \dfrac{d}{dy}[M] \not = \dfrac{d}{dx}[N]$ Therefore Differential equation is NOT exact...yet, I still need to find the integrating factor.

I need to find an "integrating factor" using this special case:

$\displaystyle u(x,y)=ax+by$

$\displaystyle \mu=\mu(ax+by)$ if an only if $\displaystyle \dfrac{M_{y}-N_{x}}{aN-bM}=R(u)=\dfrac{\mu'(u)}{\mu(u)}$

$\displaystyle \dfrac{M_{y}-N_{x}}{aN-bM}$ should simpify down to 1 (i think, professor didnt say for this special case ) by making $\displaystyle a$ and $\displaystyle b$ appropriote constant values. Here is what I get:

$\displaystyle \dfrac{-4x+8y}{(3a-10b)x^2+(4a-2b)xy+(5a-6b)y^2}$

I cannot make $\displaystyle a$ and $\displaystyle b$ values that will simplify this fraction down to "1" unless they are 0, which does not work.

I need to find the "integrating factor" but messed up or am doing it wrong, in either case any help would be appreciated. Thanks!