Finding integrating factors by inspection

[david.a491]

Hello everyone, I'm new to the forum. This is the first time I take this course and sadly I couldn't attend the first classes. I understand separable equations, but I'm struggling a bit with the following:
My teacher follows a "cheat table" (that sadly I don't have with me, so if anyone is able to figure it out, please let me know) with some common forms for differential equations and by following the table and looking at the coefficients he is able to find the proper integration factor u(x,y) without doing much work

An equation like this:

5ydx - 4xdy = xy²dy turned out in simple steps after finding the integration factor (no need to solve it, but I'd like some hints on how to find the factor

Same for this one:
(x²+2xy+y²)dx+(x²+2xy+9y²)dy=0
I worked a little on it and got the following:

(x²+2xy+y²)dx+(x²+2xy+y²+8y²)dy=0

(x²+2xy+y²)dx+(x²+2xy+y²)dy+8y²dy=0

(x²+2xy+y²)(dx+dy)+8y²dy=0

At that point I got stuck because I have no clue what to do with the (dx+dy) but according to my teacher that is also a common form and we can integrate it instantly

Thanks for any help in advance!

 

[david.a491]

Edit:

For the first problem I could divide by xy everything, and from there everything went smooth

For the second problem I found that I could factor
(x+y)²(dx+dy)+8y²dy=0

So esentially we have
(x+y)²d(x+y)+8y²dy=0
And from there I got it, I can integrate everything, but I'm still curious about how my teacher does it, he really follows a table where he uses the coefficients of the functions and is able to set up the integration factors by turning them into exponents (with some operations between them, mostly substractions I believe)

 

[david.a491]

Edit #3
I think I figured it out, the integration factor I was looking for is (x^p-1y^q-1)
It works for D.E that have the form pydx+qxdy=f(x,y) where "p" and "q" are just coefficients and the differential will turn to d((x^py^q))

If we take the equation:

5ydx - 4xdy = xy²dy

p=5, and q=-4 the integration factor is then (x⁴y^-5)
Then we multiply the whole equation by it to get

5y^-4x⁴dx-4x⁵y^-5dy=x⁵y^-3dy

d(x⁵y^-4) = x⁵y^-3dy

d(x⁵y^-4) = (x⁵y^-4)ydy

From this I divide both sides by (x⁵y^-4) and we integrate to get

ln(x⁵y^-4)+C = y²/2

Which matches the answer I got when I simply divided by xy!!

I edited this in case anyone was wanting to know what the integration factor I mentioned was, is there a way to mark the problem as solved? Thanks for everyone who read and tried to figured it out.