(6x-2y-7)/(3y+2x+4)=dy/dx needs to be progressively reduced so that we can separate variables then integrate.
FIRST SUBSTITUTION
Let x=X+h and y=Y+k where h and k are constant displacements of the x and y axes. These displacements will enable the constants -7 and 4 to be removed so that we have a homogeneous expression in X and Y.
Substituting for x and y:
dY/dX=dy/dx=(6X+6h-2Y-2k-7)/(3Y+3k+2X+2h+4).
To remove the constants: 6h-2k=7 and 3k+2h=-4, which can be written 6h+9k=-12. Subtracting to eliminate h we get: 11k=-19, so k=-19/11 and h=(-4-3k)/2=(-4+57/11)/2=13/22.
Therefore dY/dX=(6X-2Y)/(3Y+2X).
SECOND SUBSTITUTION
Let Y=vX, then dY/dX=(6X-2vX)/(3vX+2X)=(6-2v)/(3v+2). dY/dX=v+Xdv/dX, where v=v(X), a function of X. Therefore, Xdv/dX=(6-2v)/(3v+2)-v=(6-2v-3v²-2v)/(3v+2). The variables can be separated and integration is possible:
∫dX/X=∫((3v+2)/(6-4v-3v²))∫(-½)((-4-6v)/(6-4v-3v²))dv=-½ln|6-4v-3v²|.
So -2∫dX/X=ln|6-4v-3v²|. Therefore ln(a/X²)=ln|6-4v-3v²|, and a/X²=6-4v-3v², where a is the constant of integration. We can now substitute back to x and y. a/X²=6-4Y/X-3(Y/X)², a=6X²-4XY-3Y²=6(x-13/22)²-4(x-13/22)(y+19/11)-3(y+19/11)².
This evaluates to:
6x²-4xy-3y²-14x-8y=C where C is a constant.
CHECK
Differentiating we get:
12x-4xdy/dx-4y-6ydy/dx-14-8dy/dx=0.
12x-4y-14=(dy/dx)(4x+6y+8).
Divide through by 2:
6x-2y-7=(dy/dx)(2x+3y+4), (6x-2y-7)/(3y+2x+4)=dy/dx, the original DE.