Question

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

• A. \( \frac{1}{\log_e (4) - \log_e (3)} \)
• B. \( \frac{2}{\log_e (3) - \log_e (4)} \)
• C. \( - \frac{1}{\log_e (4)} \)
• D. \( \frac{1}{\log_e (3) - \log_e (4)} \)

Hint:

When solving differential equations with trigonometric and logarithmic terms, try to separate variables and apply integration techniques to solve for the unknown function.

Answer 1

The Correct Option is A

Step 1: Separate variables \( dy \) and \( dx \) to establish a relationship between \( y \) and \( x \) for the differential equation. This allows for simplification and subsequent integration.
Step 2: Apply integration methods, including substitution and integration by parts, to derive the general solution \( y(x) \). 
Step 3: Utilize the initial condition \( y \left( \frac{\pi}{4} \right) = -1 \) to ascertain the constant of integration. 
Step 4: Substitute \( x = \frac{\pi}{6} \) into the obtained solution to calculate \( y \left( \frac{\pi}{6} \right) \). The result is \( \frac{1}{\log_e (4) - \log_e (3)} \), confirming answer (1).