Question

Let \( y = y(x) \) be the solution of the differential equation \[ \cos(x \log(\cos x))^2 \, dy + (\sin x - 3 \sin x \log(\cos x)) \, dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right) \] If \( y\left( \frac{\pi}{4} \right) = -1 \), then \( y\left( \frac{\pi}{6} \right) \) is equal to:

• A. \( 1 - \log 4 \)
• B. \( 2 \log 3 - \log 4 \)
• C. \( -1 \log 4 \)
• D. \( 1 \log 3 - \log 4 \)

Hint:

To solve differential equations, check if separation of variables or an integrating factor is useful.

Answer 1

The Correct Option is A

We solve the differential equation using standard methods of solving first-order differential equations. The solution will involve the integral of the equation, and after applying the initial condition \( y\left( \frac{\pi}{4} \right) = -1 \), we find that: \[ y\left( \frac{\pi}{6} \right) = 1 - \log 4. \]