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**Differential equation first, first degree help!!**

## Homework Statement

solve: x.dy + y.dx + (x.dy - y.dx)/(x^2 + y^2) = 0

## Homework Equations

if M.dx + N.dy = 0 has to be exact then

equation 1: partial derivative of M w.r.t y (keeping x constant) = partial derivative of N w.r.t x (keeping y constant)

## The Attempt at a Solution

the idea is to find if this equation is exact because once you do that the integration is easy..

but

now i first i simplify the equation by multiplying throughout by (x^2 + y^2)

simplified form: (x^3 + x.y^2 - y).dx + (y^3 + y.x^2 + x).dy = 0

try to check if it satisfies equation 1: partial derivative of M w.r.t y (keeping x constant) = partial derivative of N w.r.t x (keeping y constant)

here M = (x^3 + x.y^2 - y) & N = (y^3 + y.x^2 + x)

we get 2.x.y - 1 is not equal to 2.x.y + 1

now i try another method i.e. group the terms without multiplying throughout by (x^2 + y^2)

simplified equation: [x - (y/(x^2 + y^2))].dx + [y + (x/(x^2 + y^2))].dy = 0

now when we apply equation 1 criteria we get:

partial derivative of M w.r.t y (keeping x constant) = (y^2 - x^2)/(x^2 + y^2)^2 = partial derivative of N w.r.t x (keeping y constant) = (y^2 - x^2)/(x^2 + y^2)^2

my question is - why is it that i am getting the two methods to be different???