If \(y = y(x)\), \(-\displaystyle\frac{\pi}{2} \lt y \lt \frac{\pi}{2}\), is the solution of the differential equation \(\cos y\displaystyle\frac{dy}{dx} = e^{x}\) satisfying \(y(0) = 0\), then \(y\left(\log_e \displaystyle\frac{3}{2}\right)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{6}\)
The variables are already separated:
\( \cos y\,dy = e^{x}\,dx \)
Integrating both sides:
\( \sin y = e^{x} + C \)
Using \(y(0) = 0\):
\( 0 = 1 + C \), so \( C = -1 \)
The particular solution is
\( \sin y = e^{x} - 1 \)
At \(x = \log_e \frac{3}{2}\), \(e^{x} = \frac{3}{2}\):
\( \displaystyle \sin y = \frac{3}{2} - 1 = \frac{1}{2} \)
Since \(-\frac{\pi}{2} \lt y \lt \frac{\pi}{2}\),
\( \displaystyle y = \frac{\pi}{6} \)