Question

For the differential equation $xy\frac{dy}{dx}=(x+2)(y+2)$,find the equation of curve passing through the point $(1,-1)$.

Answer 1

$xy\frac{dy}{dx}=(x+2)(y+2)$
$\frac{y}{y+2}dy=\frac{x+2}{x}dx\Rightarrow \int 1-\frac{2}{2+y}=\int (\frac{2}{x}+1)dx$
$y-2\log |y+2|=2\log x+x+C$
but passes through $(1.-1),x=1,y=-1$
$-1-2\log 1=2\log 1+1+C$ $\therefore C=-2$