The general solution of the differential equation \( \displaystyle\frac{dy}{dx}=2^{-y} \) is:
(where \( c \) is an arbitrary constant )
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 2^y=x\log_e 2+c \)
Since \( 2^{-y}=\frac{1}{2^y} \), the variables separate:
\( \displaystyle \frac{dy}{dx}=\frac{1}{2^y} \)
\( 2^y\,dy=dx \)
Using \( \int a^y\,dy=\frac{a^y}{\log_e a} \):
\( \displaystyle \frac{2^y}{\log_e 2}=x+c_1 \)
\( 2^y=x\log_e 2+c_1\log_e 2 \)
Writing the constant \( c_1\log_e 2 \) as c:
\( 2^y=x\log_e 2+c \)
Shortcut: Differentiate Option (3) with respect to x: \( 2^y\log_e 2\,\frac{dy}{dx}=\log_e 2 \), so \( \frac{dy}{dx}=\frac{1}{2^y}=2^{-y} \).