If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \cos^2 y\) satisfying \(y(0) = \displaystyle\frac{\pi}{4}\), then \(y\left(\sqrt{3} - 1\right)\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{\pi}{3}\)
The right side has only y, so the variables separate by dividing by \(\cos^2 y\):
\( \displaystyle \sec^2 y\,dy = dx \)
Integrating both sides:
\(\tan y = x + C\)
Putting x = 0, \(y = \frac{\pi}{4}\):
\(1 = 0 + C,\quad C = 1\)
\(\tan y = x + 1\)
At \(x = \sqrt{3} - 1\):
\(\tan y = \sqrt{3}\)
The solution starts at \(\frac{\pi}{4}\) and \(\tan y\) stays finite, so y stays between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). Hence
\( \displaystyle y = \frac{\pi}{3} \)