Question

Solve the differential equation :
$\frac{dy}{dx}=-\frac{2x+y+1}{4x+2y-1}$

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

Answer 1

The correct option is A $\log (2x+y-1)+x+2y=c$

$\frac{dy}{dx}=-\frac{2x+y+1}{4x+2y-1}$
Substitute $2x+y=t\Rightarrow 2+\frac{dy}{dx}=\frac{dt}{dx}$
$\frac{dt}{dx}-2=-\frac{t+1}{2t-1}\Rightarrow \frac{dt}{dx}=\frac{3(t-1)}{2t-1}\Rightarrow \frac{dt}{dx}\frac{2t-1}{3(t-1)}=1\Rightarrow \int dt\frac{2t-1}{3(t-1)}=\int dx\Rightarrow \log (2x+y-1)+x+2y=c$