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Verify that: $$\frac12(Mx+Ny)d(\ln(xy))+\frac12(Mx-Ny)d(\ln(x/y))=Mdx+Ndy$$

Hence show that, if the de $Mdx+Ndy=0$ is homogenous, then $Mx+Ny$ is an integrating factor unless $Mx+Ny=0$

Note: Verification is trivial, hence nothing much to be done there, but I couldnt solve the second part of the question "Hence..." so for the completeness of the problem I added it. Further on, isnt the statement " $Mdx+Ndy=0$ is homogenous " superfluous as RHS is already zero, so why add the word homogenous. Perhaps I am being pedantic? And lastly I would like to have some hints in solving the INTEGRATING Factor part.

EDIT: My approach I approached like this: I multiplied the function $Mx+Ny$ to both sides of the equation $Mdx+Ndy=0$ and tried to show, that $d(u(x,y))=0$ but I couldnt prove it.

Soham

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    I think that in this context "homogeneous" might mean that $M$ and $N$ are homogeneous polynomials in $x$ and $y$, and of the same degree. E.g., $$(2x^2+3xy+4y^2)dx+(5x^2-6xy-7y^2)dy=0$$2012-06-14
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    aah.. I see....2012-06-15

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