The general solution of the given differential equation \( dy+(2y-e^{-2x})dx=0 \), is:
[where c is an arbitrary constant]
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( e^{2x}y=x+c \)
Dividing by dx, the equation is linear in y:
\( \displaystyle \frac{dy}{dx}+2y=e^{-2x} \)
The integrating factor is \( e^{\int 2\,dx}=e^{2x} \), so
\( \displaystyle y\,e^{2x}=\int e^{-2x}\cdot e^{2x}\,dx \)
\( \displaystyle =\int 1\,dx \)
\( \displaystyle =x+c \)