Question

Solve:
$(x+y-1)dx+(2x+2y-3)dy=0$

Answer 1

$-(x+y-1)dx=(2x+2y-3)dy$

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

Let $x+y=t$

$1+\frac{dy}{dx}=\frac{dt}{dx}$

(On differentiating wrt n)

$\Rightarrow -\frac{(t-1)}{(2t-3)}=\frac{dt}{dx}-1$

$\Rightarrow 1-\frac{t-1}{2t-3}=\frac{dt}{dx}$

$\Rightarrow \frac{dt}{dx}=\frac{2t-3-t+1}{2t-3}$

$=\frac{t-2}{2t-3}=\frac{t-2}{2(t-\frac{3}{2})}$

$\Rightarrow 2\frac{dt}{dx}=\frac{t-2}{t-\frac{3}{2}}$

$\int \frac{(2t-3)}{(t-2)}dt=\int dx$

$\int \left(\frac{2t-4+1}{t-2}\right)dt=\int dx$

$\Rightarrow \int (\frac{2t-4}{t-2}+\frac{1}{t-2})dt=\int dx$

$\Rightarrow \int 2dt+\int \frac{dt}{t-2}=\int dx$

$\Rightarrow 2t+ln|t-2|=x+c$

$\Rightarrow 2(x+y)+ln|x+y-2|=x+c$