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

$\frac{ydy}{y+2}=\left(\frac{x+2}{x}\right)dx$

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

Integrating both sides $\Rightarrow$

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

$y-2ln(y+2)=x+2lnx+c$

$(1,-1)$ satisfies this equation

$(-1)-2ln(-1+2)=1+2ln1+C$

$-1-0=1+0+c$

$-2=c$

$\therefore$ equation is

$y-2ln(y+2)=x+2lnx-2$