If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = y\tan x\), \(-\frac{\pi}{2} \lt x \lt \frac{\pi}{2}\), satisfying \(y(0) = 2\), then \(y\left(\frac{\pi}{3}\right)\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4
The equation is variables separable:
\( \displaystyle \frac{dy}{y} = \tan x\,dx \)
Integrating both sides, with \(\int \tan x\,dx = \log|\sec x|\):
\(\log|y| = \log|\sec x| + \log|C|\)
\(y = C\sec x\)
Putting x = 0, y = 2:
\(2 = C\sec 0 = C\)
\(y = 2\sec x\)
At \(x = \frac{\pi}{3}\), where \(\sec\frac{\pi}{3} = 2\):
\( \displaystyle y\left(\frac{\pi}{3}\right) = 2 \times 2 = 4 \)