Undergrad How to find the integrating factor? (1st order ODE)

[hellotheworld]

x² + y + y²dx - x dy = 0

Integrating factor, I(x,y) = -1 / (x² + y²)

How to find the integrating factor ?Why I cannot use below method to solve the ode ?

(1/N)(M_y - N_x) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR
(1/M)(M_y - N_x) = h(y) , I(x,y)=exp( -∫ h(y) dy)

 

[haruspex]

x² + y + y²dx - x dy = 0

That equation makes no sense. You cannot mix infinitesimals and non-infinitesimals like that.

 

[Mark44]

x² + y + y²dx - x dy = 0

Did you mean $(x^2 + y + y^2)dx - x dy = 0$?

Integrating factor, I(x,y) = -1 / (x² + y²)

How to find the integrating factor ?Why I cannot use below method to solve the ode ?

(1/N)(M_y - N_x) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR
(1/M)(M_y - N_x) = h(y) , I(x,y)=exp( -∫ h(y) dy)