Solve `(dy)/(dx)=((x+y)^2)/((x+2)(y-2))`

Explanation

`(dy)/(dx) = (x+y)^(2)/((x+2)(y-2))`
On putting `X=x+2` and `Y=y-2`, the given differential equation reduces to
`(dY)/(dX) = (X+Y)^(2)/(XY)`
Put `Y=VX`, i.e., `(dY)/(dX) = V+X(dV)/(dX)`
`therefore V+X(dV)/(dX) = (1+V)^(2)/(V)`
or `V/(2V+1)dV= (dX)/(X)`
or `int1-1/(1+2V)dV=2int(dX)/(X)`
or `V-1/2"ln"(1+2V)=2"ln"X+C`
or `X^(4)(1+(2Y)/(X))=Ce^(2Y//X)`
Where, `X=x+2 and `Y=y-2`.