Question

$\frac{dy}{dx}=\frac{x+y+1}{2x+2y+3}$

Answer 1

Given,

$\frac{dy}{dx}=\frac{x+y+1}{2x+2y+3}$

$\frac{dy}{dx}=\frac{x+y+1}{2(x+y)+3}$

substitute $x+y=v\to 1+\frac{dy}{dx}=\frac{dv}{dx}$

$\frac{dv}{dx}-1=\frac{v+1}{2v+3}$

$\frac{dv}{dx}=\frac{3v+4}{2v+3}$

$\frac{2v+3}{3v+4}dv=dx$

integrating on both sides, we get,

$\int \frac{2v+3}{3v+4}dv=\int dx$

$\frac{2}{3}v+\frac{1}{9}\log |3v+4|=x+c$

$\frac{2}{3}(x+y)+\frac{1}{9}\log |3(x+y)+4|=x+c$