Question

The solution of the differential equation $\frac{dy}{dx}=\frac{x-y+3}{2(x-y)+5}$ is

• A. $2(x-y)+\log (x-y)=x+c$
• B. $2(x-y)-\log (x-y+2)=x+c$
• C. $2(x-y)+\log (x-y+2)=x+c$
• D. None of the above

Answer 1

The correct option is C $2(x-y)+\log (x-y+2)=x+c$

Given differential equation is

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

Let $x-y=v\Rightarrow \frac{dy}{dx}=1-\frac{dv}{dx}$

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

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

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

$\Rightarrow 2v+\log (v+2)=x+c$

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