Solving ODE: Integrating Factor for Problem 4d

[roryhand]

Howdy, I've read this forum for some time, however this is my first post. I am attempting to solve this ODE. I am looking to find an integrating factor, then solve. I have attached the link to the problem set if my input here is ambiguous. Number 4d. Thank you kindly for any help you might lend.


(2x^2)+y+((x^2)*y)-x)dy/dx=0

My reasoning takes me as far as the integrating factor being exp(int( ? )dx)

https://people.creighton.edu/%7Elwn70714/DE_Assignments/DE%20Assignment%202%20PDF.pdf

 

[Emieno]

(2x^2)+y+((x^2)*y)-x)dy/dx=0

(2x^2+y)dx + [(x^2)*y-x]dy=0

Now you just need re-read your text-book about how to solve Pdx+Qdy=0, after checking some conditions on P&Q if such an equation has roots or none.

 

[saltydog]

Or can approach it this way:

$$(2x^2+y)dx+(x^2y-x)dy=0$$

So, after a quick check for homogeneous, exact, and explicit calc. of an integrating factor via partials, we expand the differentials and attempt to group them together to form exact differentials:

$$2x^2dx+ydx+x^2ydy-xdy=0$$

Well, the ydx-xdy can be grouped as:

$$y^2\left(\frac{ydx-xdy}{y^2}\right)$$

This leaves us with:

$$2x^2dx+x^2ydy+y^2 d\left(\frac{x}{y}\right)$$

Can you re-arrange this now to obtain exact differentials which can be integrated?