Question

The solution of the differential equation $\frac{dy}{dx}=\frac{x-2y+1}{2x-4y}$

• A. ${(x-2y)}^2-2x=C$
• B. ${(x-2y)}^2+x=C$
• C. $(x-2y)+2x^2=C$
• D. $(x-2y)+x^2=C$

Answer 1

The correct option is A ${(x-2y)}^2-2x=C$

Let $x-2y=t\Rightarrow 1-2\frac{dy}{dx}=\frac{dt}{dx}$

$\Rightarrow \frac{1}{2}(1+\frac{dt}{dx})=\frac{t+1}{2t}$

$\Rightarrow \frac{dt}{dx}=\frac{1}{t}$

Integrate on both sides

$\Rightarrow \frac{t^2}{2}=x+c$

$\Rightarrow (x-2y{)}^2=2x+c$