# Differential equation first, first degree help

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???

## Answers and Replies

eh... anyone???? wold really help.. this has been bugging me...

Have you seen integrating factors yet? That whole method is based on the idea of multiplying the equation by a function to put it in a 'nice' form. When you have an equation M(x,y)dx + N(x,y)dy=0, multiplying by anything is going to change the derivatives. It's very likely that it will change the partial derivatives of M and N in different ways, which may or may not help you make the equation into an exact equation. For example, if I multiply the equation by x, that's going to change the x partial derivative of N, but it's not going to change the y partial of M.

SammyS
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Science Advisor
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## 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 ...
Notice that, d(xy) = x.dy + y.dx .

Also, d(y/x) = (xdy - ydx)/x2

So, you can rewrite your equation as: (x2 + y2)d(xy) + x2(d(y/x)) = 0

This suggests that you let u = xy and v = x/y.

Can you take it from here?