The solution of the equation \(\frac{dy}{dx}=3^{y-x}\) is

\(\frac{1}{3^{x}}-\frac{1}{3^{y}}=c\)

The correct answer is Option (4) → \(\frac{1}{3^{x}}-\frac{1}{3^{y}}=c\)

\(\frac{dy}{dx}=3^{y-x}\)

$⇒\int 3^{-y}dy=\int 3^{-x}dx$

$⇒-\frac{3^{-y}}{\log_e3}=-\frac{3^{-x}}{\log_e3}+C$

$⇒\frac{3^{-x}}{\log_e3}-\frac{3^{-y}}{\log_e3}=C$  [$\log 3 ×C$ = Constant]

$⇒\frac{1}{3^{x}}-\frac{1}{3^{y}}=C$