General solution of the differential equation \( \displaystyle\frac{dy}{dx}=e^{ax+by} \) is: (where C is an arbitrary constant)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( be^{ax}+ae^{-by}=c \)
Since \( e^{ax+by}=e^{ax}\,e^{by} \), the variables separate:
\( \displaystyle \frac{dy}{dx}=e^{ax}e^{by} \)
\( \displaystyle e^{-by}\,dy=e^{ax}\,dx \)
Integrating both sides:
\( \displaystyle -\frac{e^{-by}}{b}=\frac{e^{ax}}{a}+k \)
Multiplying by ab and moving the terms to one side:
\( \displaystyle -ae^{-by}=be^{ax}+abk \)
\( \displaystyle be^{ax}+ae^{-by}=-abk=c \)
Shortcut: Differentiate Option (3): \( abe^{ax}-abe^{-by}\frac{dy}{dx}=0 \), so \( \frac{dy}{dx}=\frac{e^{ax}}{e^{-by}}=e^{ax+by} \), which is the given equation.
Why the other options are incorrect
Option (1) \( e^{ax}+e^{-by}=c \) — Incorrect. Differentiating gives \( \frac{dy}{dx}=\frac{a}{b}e^{ax+by} \); the factors a and b from the integration are missing.
Option (2) \( e^{ax+by}=c \) — Incorrect. Differentiating gives \( \frac{dy}{dx}=-\frac{a}{b} \), a constant, so this is not a solution.
Option (4) \( ae^{ax}-be^{-by}=c \) — Incorrect. Differentiating gives \( \frac{dy}{dx}=-\frac{a^2}{b^2}e^{ax+by} \); the coefficients are interchanged and the sign is wrong.