The general solution of the differential equation \(\displaystyle\frac{dy}{dx} = e^{y}\sin x\) is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(e^{-y} = \cos x + C\)
Dividing by \(e^{y}\) separates the variables:
\( e^{-y}\,dy = \sin x\,dx \)
Integrating both sides:
\( \displaystyle \int e^{-y}\,dy = \int \sin x\,dx \)
\( -e^{-y} = -\cos x + C_1 \)
Multiplying by \(-1\) and writing C for \(-C_1\):
\( e^{-y} = \cos x + C \)
Why the other options are incorrect
Option (2) \(e^{y} = \cos x + C\) — Incorrect. Differentiating gives \(e^{y}\frac{dy}{dx} = -\sin x\), that is \(\frac{dy}{dx} = -e^{-y}\sin x\).
Option (3) \(e^{-y} = -\cos x + C\) — Incorrect. Only one side was multiplied by \(-1\). Differentiating gives \(-e^{-y}\frac{dy}{dx} = \sin x\), that is \(\frac{dy}{dx} = -e^{y}\sin x\).
Option (4) \(e^{-y} = \sin x + C\) — Incorrect. Differentiating gives \(\frac{dy}{dx} = -e^{y}\cos x\); the integral of \(\sin x\) is \(-\cos x\), not \(\sin x\).