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## Homework Statement

By means of substitution x=X+1, y=Y+2 ,shwo that the equation dy/dx=(2x-y)/(x+2y+5) can be reduced to dY/dX=(2X-Y)/(X+2Y).Hence, find the general solution of the given equation.

## Homework Equations

## The Attempt at a Solution

The first part is quite simple to prove.

Second part,

since its a homogenous differential equation, i would use the substitution Y=vX here.

v+X(dv/dX)=(2X-vX)/(2vX+X)

[tex]-\frac{1}{2}\int \frac{4v+2}{2v^2+2v+2} dv=\int \frac{1}{X} dx[/tex]

ln |2v^2+2v+2|=-2ln |X|-c

then v=Y/X

ln |(2Y^2)/X^2+2Y/X+2|=-2ln |X|-c'

Then substitute back again from the first part,

the solution is

ln |2(y-2)^2/(x-1)^2+2(y-2)/(x-1)+2|=-2ln |x-1|-c'

AM i correct?