The general solution of the differential equation \(\displaystyle\frac{dy}{dx}=e^{ax+by}\) is: (Here C is an arbitrary constant)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(ae^{-by}+be^{ax}=C\)
Since \(e^{ax+by} = e^{ax}e^{by}\), the variables separate:
\(e^{-by}\,dy = e^{ax}\,dx\)
Integrating both sides:
\( \displaystyle -\frac{1}{b}e^{-by} = \frac{1}{a}e^{ax} + C_1 \)
Multiplying by \(-ab\):
\(ae^{-by} = -be^{ax} - abC_1\)
\(ae^{-by} + be^{ax} = C\), where \(C = -abC_1\)