Question

Solve:
$\frac{dy}{dx}=\frac{(x-y)+3}{2(x-y)+5}$

Answer 1

We have,

$\frac{dy}{dx}=\frac{\left(x-y\right)+3}{2\left(x-y\right)5}$

Let $x-y=t$

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

Therefore,

$1-\frac{dt}{dx}=\frac{t+3}{2t+5}1-\frac{\left(t+3\right)}{2t+5}=\frac{dt}{dx}\frac{2t+5-t-3}{2t+5}=\frac{dt}{dx}\frac{t+2}{2t+5}=\frac{dt}{dx}dx=\frac{2t+5}{t+2}dtdx=\left[\frac{2\left(t+2\right)+1}{t+2}\right]dt$

On taking integration both sides, we get

$\int dx=\int 2dt+\int \frac{1}{t+2}+Cx=2t+\ln (t+2)+Cx=2\left(x-y\right)+\ln (x-y+2)+C{C}^{\prime }=x-2y+\ln (x-y+2)$

Hence, this is the answer.