Question

Solution of differential equation

$\frac{dy}{dx}=\frac{(x-y)+3}{2(x-y)+5}$ is:
(where $c$ is integration constant and $\log$ is given with base ${}^{\prime }{e}^{\prime }$)

• A. $c=x-2y-\log |x-y+1|$
• B. $c=x-y+\log |x-y+2|$
• C. $c=2x+2y+\log |x-y+1|$
• D. $c=x-2y+\log |x-y+2|$

Answer 1

The correct option is D $c=x-2y+\log |x-y+2|$

Put $x-y=t$ in the given
differential equation such that
$1-\frac{dy}{dx}=\frac{dt}{dx}$

$\Rightarrow 1-\frac{dt}{dx}=\frac{t+3}{2t+5}$ $\Rightarrow \frac{dt}{dx}=1-\frac{t+3}{2t+5}=\frac{t+2}{2t+5}$
$\Rightarrow \int dx=\int \frac{2t+5}{t+2}dt=2\int dt+\int \frac{dt}{t+2}$
$\Rightarrow x+c=2t+\log |t+2|\Rightarrow x+c=2(x-y)+\log \left|(x-y+2)\right|$

$\therefore c=x-2y+\log |x-y+2|$