For the differential equation `x y(dy)/(dx)=(x+2)(y+2)`, find the solution curve passing through the point `(1, 1)`.

Explanation

Given, differential equation,
`xy(dy)/(dx)=(x+2)(y+2)`…….`(1)`
`implies (y)/(y+2)dy=(x+2)/(x)dx`
`implies int(y)/(y+2)dy=int(x+2)/(x)dx`
`implies int((y+2-2)/(y+2))dy=int(x+2)/(x)dx`
`implies int(y+2)/(y+2)dy-int(2)/(y+2)dy=int(x)/(x)dx+int(2)/(x)dx`
`impliesintdy-2int(1)/(y+2)dy=intdx+2int(1)/(x)dx`
`implies y-2log|y+2|=x+2log|x|+C`..........`(2)`
Curve passes through the point `(1,-1)` , so put `x=1` and `y=-1`,
`-1-2log|-1+2|=1+2log1+C`
`implies -1-2log1=1+2log1+CimpliesC=-2`
Therefore, from equation `(2)`,
`y-2log|y+2|=x+2log|x|-2`
`implies y-x+2=2log|x(y+2)|`
`implies y-x+2=log[x^(2)(y+2)^(2)]`
which is the equation of required curve.