Behold, MAT244 in flowchart form!

[Daniel Clark]

So I had this idea of putting all of the lecture material of MAT244 in a giant flowchart. So I did.

A few notes:
Dotted lines means the question is reduced to the smaller one

For the partial derivatives I used notation not used in class, the curly "d" with a subscript just means a derivative with respect to the variable in the subscript

I tried to keep it on one page and have all the notation as general as possible, which might have made it nigh impossible to read.  Sorry about that.

I don't have EVERYTHING, and I'm planning to update it as the class goes on.  Constrictive criticism welcome.

[Victor Ivrii]

Wowzers!!!

But you forgot in integrating factor case of $\mu= \mu (xy)$.

And somehow relevant
https://www.nytimes.com/2010/04/27/world/27powerpoint.html?hp

[Kathryn Bucci]

I think that if you had a non-exact homogeneous equation and you can't easily find an integrating factor, you could also solve it by making a substitution y=vx.. then it would be reduced to a separable equation.

[Daniel Clark]

RE: Kathryn Bucci
Incorrect, counterexample x +(x^2-y^2)y' = 0
using the substitution gives us x +(x^2-v^2x^2)v' =0 which is arguably worse. 
That would work if the derivative is a function of the ratio, which i covered in my unholy diagram on the left. 

[Victor Ivrii]

 Daniel, Kathryn

The word "homogeneous" is overused. Kathryn is right if equation is homogeneous which means $y'=f(y/x)$. Daniel meanwhile thinks about linear homogeneous, which is a completely different story.

[Kathryn Bucci]

Daniel, Kathryn

The word "homogeneous" is overused. Kathryn is right if equation is homogeneous which means $y'=f(y/x)$. Daniel meanwhile thinks about linear homogeneous, which is a completely different story.

Yes that is what I was referring to - sorry, I should have been more clear.