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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 = eln 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.