Question

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

Answer 1

We have,
$$\frac{dy}{dx}=\frac{(x-y)+3}{2(x-y)+5}$$

Putting x - y = v

$$\Rightarrow 1-\frac{dy}{dx}=\frac{dv}{dx}$$

$$\Rightarrow \frac{dy}{dx}=1-\frac{dv}{dx}$$

$$\therefore 1-\frac{dv}{dx}=\frac{v+3}{2v+5}$$

$$\Rightarrow \frac{dv}{dx}=1-\frac{v+3}{2v+5}$$

$$\Rightarrow \frac{dv}{dx}=\frac{2v+5-v-3}{2v+5}$$

$$\Rightarrow \frac{dv}{dx}=\frac{v+2}{2v+5}$$

$$\Rightarrow \frac{2v+5}{v+2}dv=dx$$

Integrating both sides, we get

$$\int \frac{2v+5}{v+2}dv=\int dx$$

$$\Rightarrow \int \frac{2v+4+1}{v+2}dv=\int dx$$

$$\Rightarrow \int (\frac{2v+4}{v+2}+\frac{1}{v+2})dv=\int dx$$

$$\Rightarrow 2\int dv+\int \frac{1}{v+2}dv=\int dx$$

$$\Rightarrow 2v+\log |v+2|=x+C$$

$$\Rightarrow 2(x-y)+\log |x-y+2|=x+C$$