# Thread: [SOLVED] Help with &quot;Exact Differential Equations&quot;

1. ## [SOLVED] Help with &quot;Exact Differential Equations&quot;

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!

2. Hey, isn't it homogeneous? Why not solve it via $\displaystyle y=vx$?

3. That would make this problem a lot easier, except I dont know how to do a problem similar to this, where I need to find an integrating factor in the form $\displaystyle ax+by$, to make the equation exact.