The general solution of the differential equation \(\log_e \left(\displaystyle\frac{dy}{dx}\right) = ax + by\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{e^{ax}}{a} + \frac{e^{-by}}{b} + C = 0\), where C is constant of integration
Removing the logarithm gives an equation in variables separable form:
\( \displaystyle \frac{dy}{dx} = e^{ax + by} = e^{ax}\,e^{by} \)
\(e^{-by}\,dy = e^{ax}\,dx\)
Integrating both sides:
\( \displaystyle \int e^{-by}\,dy = \int e^{ax}\,dx \)
\( \displaystyle -\frac{e^{-by}}{b} = \frac{e^{ax}}{a} + C \)
Rearranging:
\( \displaystyle \frac{e^{ax}}{a} + \frac{e^{-by}}{b} + C = 0 \)