If \(y = y(x)\), \(0 \le x \lt \displaystyle\frac{\pi}{2}\), is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = y\tan x\) with \(y(0) = 2\), then \(y\left(\displaystyle\frac{\pi}{3}\right)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4
Separating the variables:
\( \displaystyle \frac{dy}{y} = \tan x\,dx \)
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, with \(\sec 0 = 1\):
\(C = 2\)
\(y = 2\sec x\)
At \(x = \frac{\pi}{3}\), \(\sec\frac{\pi}{3} = 2\):
\(y = 2 \times 2 = 4\)
Why the other options are incorrect
Option (1) 1 — Incorrect. This comes from \(y = 2\cos x\), which takes \(\int \tan x\,dx\) as \(\log|\cos x|\); the correct integral is \(-\log|\cos x| = \log|\sec x|\).
Option (3) 8 — Incorrect. This is \(2\sec^2\frac{\pi}{3}\); the solution has \(\sec x\), not \(\sec^2 x\).
Option (4) \(\sqrt{3}\) — Incorrect. This is \(\tan\frac{\pi}{3}\), not the value of \(2\sec\frac{\pi}{3}\).