The general solution of the differential equation \( \log\left(\displaystyle\frac{dy}{dx}\right)=ax+by \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+c \), where \( c \) is an arbitrary constant
Since \( \log\left(\frac{dy}{dx}\right)=ax+by \),
\( \displaystyle \frac{dy}{dx}=e^{ax+by}=e^{ax}e^{by} \)
The variables separate:
\( \displaystyle e^{-by}\,dy=e^{ax}\,dx \)
Integrating both sides,
\( \displaystyle \frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+c \)