The solution of the differential equation \( \displaystyle\frac{dy}{dx}=2^{x+y} \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 2^{x}+2^{-y}=c \) where c is an arbitrary constant
Since \( 2^{x+y}=2^{x}\cdot 2^{y} \), the variables separate:
\( \displaystyle \frac{dy}{dx}=2^{x}\cdot 2^{y} \)
\( \displaystyle 2^{-y}\,dy=2^{x}\,dx \)
Integrating both sides, with \( \int a^{u}\,du=\frac{a^{u}}{\log_e a} \):
\( \displaystyle -\frac{2^{-y}}{\log_e 2}=\frac{2^{x}}{\log_e 2}+k \)
Multiplying by \( \log_e 2 \) and rearranging:
\( \displaystyle 2^{x}+2^{-y}=-k\log_e 2 \)
\( \displaystyle 2^{x}+2^{-y}=c \)