(adsbygoogle = window.adsbygoogle || []).push({}); Easier way of finding Integrating Factor for Exact Differential Equation??

1. The problem statement, all variables and given/known data

,find an integrating factor and then solve the following:

[4(x^{3}/y^{2})+(3/y)]dx + [3(x/y^{2})+4y]dy = 0

2. Relevant equations

u(y)=y^{2}is a valid integrating factor that yields a solution x^{4}+3xy+y^{4}=c

,no clue how though!

3. The attempt at a solution

I understand exact differential equations to really be a somewhat perverted form of a gradient of a multivariable surface defined by f=f(x,y)... and that solving an exact differential equations is comparable to finding the potential function using the fundamental theorem of line integrals.

,here with this equation I tried the usual methods

I first realized that the equation is NOT exact (M_{y}does not equal N_{x}), or in other words the two parts of the differential equation could not have possibly come from the same function f=f(x,y) in their current forms because f_{xy}=f_{yx}for any f(x,y).

So, I multiplied by an integrating factor u(x,y) such that the differential equation would be exact or such that (uM)_{y}=(uN)_{x}

,so far in my math education I've pretty much disregarded the idea of finding an integrating factor that is not solely a function of either y or x - it simply would be too hard.

,so in assuming u to be a function of either solely x or y I had hoped to find a linear and separable DE through the following:

du/dx=[u(M_{y}-N_{x}]/N or

du/dy=[u(N_{x}-M_{y}]/M

,unfortunately both of these yield seemingly impossibly, or at least algebraically disgusting results...

Any ideas on an easier or other method for solving for this/other integrating factors?

,and on a slightly related note, any good referenced here in the forums for writing better math symbols?

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# Easier way of finding Integrating Factor for Exact Differential Equation?

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