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: The given differential equation can be simplified and solved by separating variables and integrating. First, isolate \( dy \) and \( dx \) terms to obtain the relation between \( y \) and \( x \). 
Step 2: After applying the appropriate integration techniques, such as substitution and integration by parts, we get the general solution for \( y(x) \). 
Step 3: Use the initial condition \( y \left( \frac{\pi}{4} \right) = -1 \) to determine the constant of integration. 
Step 4: Finally, substitute \( x = \frac{\pi}{6} \) into the solution to get \( y \left( \frac{\pi}{6} \right) \), which evaluates to \( \frac{1}{\log_e (4) - \log_e (3)} \). Thus, the correct answer is (1).