Exact Differential Equations and Integrating Factor

[Color_of_Cyan]

Homework Statement

"Show that each of the given differential equations of the form M(x,y)dx + N(x,y)dy = 0 are
exact, and then find their general solution using integrating factors μ(x) = e^∫h(x)dx and μ(x) = e^∫g(y)dy

Homework Equations

(x² + y² + x)dx + (xy)dy = 0

The Attempt at a Solution

Can someone please tell me how to get started or what to do after? This is very confusing for me. I did THIS EXACT SAME problem before but in another way, where it said to show exactness and I had to use the integrating factor to show exactness and THEN solve using grouping. NOW it's saying to use integrating factor to find the general solution itself. I don't get it, this is very confusing to me. The ANSWER for the general solution ends up to be 3x⁴ + 6x²y² + 4x³ = C... Anyways:

(x² + y² + x)dx + (xy)dy = 0

M(x,y) = ( x² + y² + x ) ; N(x,y) = xy

h(x) = [ ( dM / dy ) - ( dN / dx ) ] / N ;

integrating factor = e^∫h(x)dx

h(x) = (2y - y) / xy = y/xy = 1/x ; h(x) = 1/x

integrating factor = e^∫(1/x)

integrating factor = e^{ln x} ; integrating factor = x

Now what do I do with it? Am I to just solve using 'grouping' or solve using 'brute force' method? Or do I take integrating factor again except using N(x,y) and then what? This is really getting me mad.

 

[ideasrule]

I'm also confused. Are you sure you're supposed to show (x^2 + y^2 + x)dx + (xy)dy = 0 is exact? But it's not. The whole point of using integrating factors is to make a non-exact equation exact.

 

[Color_of_Cyan]

I'm also confused. Are you sure you're supposed to show (x^2 + y^2 + x)dx + (xy)dy = 0 is exact? But it's not. The whole point of using integrating factors is to make a non-exact equation exact.

That's my point :/

Should I just go ask my instructor?