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

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

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

Integrating both sides, we get

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

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

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

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

Given: Curve passes through $(1,-1)$

Substituting $x=1,y=-1\text{in}\left(i\right)$,

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

$-1=1+c$

$c=-2$

Substituting $c=-2\text{in}\left(i\right)$,

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

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

$\because \log b^n=n\log b$

$y-x+2=\log \left(x^2\right(y+2^2\left)\right)$
$\because \log a+\log b=\log ab,a>0,b>0$
Hence, required equation pof the curve is $y-x+2=\log \left(x^2\right(y+2{)}^2)$