Question

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

Answer 1

Equation can be written as

$\frac{y}{(y+2)}dy=\frac{x+2}{x}dx$

$(1-\frac{2}{y+2})dy=\frac{x+2}{x}dx$

Integrating both the sides, we get

$y-2\log (y+2)=x+2\log \left(x\right)+c$

As mentioned, this curve will pass from $(1,-1)$

$-1-2\log (-1+2)=1+2\log \left(1\right)+c$

$\therefore c=-2$
Equation of the curve will be:
$y-2\log (y+2)=x+2\log \left(x\right)-2$