Question

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

Answer 1

Given,

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

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

integrating on both sides, we get,

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

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

substitute $x=1,y=-1$

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

$\Rightarrow c=-2$

$\therefore y-2\log (y+2)=x+2\log x-2$

$y-x+2=\log (y+2{)}^2+\log x^2$

$y-x+2=\log \left[x^2\right(y+2{)}^2]$