Find the general solution of the differential equation: $\log \left( \frac{dy}{dx} \right) = ax + by.$

$\frac{e^{ax}}{a} + \frac{e^{-by}}{b} = C$

The correct answer is Option (1) → $\frac{e^{ax}}{a} + \frac{e^{-by}}{b} = C$ ##

Given differential equation is

$\log \left( \frac{dy}{dx} \right) = ax + by$

$\Rightarrow \frac{dy}{dx} = e^{ax + by}$

$\Rightarrow \frac{dy}{dx} = e^{ax} \cdot e^{by}$

$\Rightarrow \frac{dy}{e^{by}} = e^{ax} dx$

$\Rightarrow e^{-by} dy = e^{ax} dx$

On integrating both sides, we get

$\int e^{-by} dy = \int e^{ax} dx$

$\frac{e^{-by}}{-b} = \frac{e^{ax}}{a} + C$

$\Rightarrow \frac{e^{ax}}{a} + \frac{e^{-by}}{b} + C = 0$