Question

If \[ \sec x \frac{dy}{dx} - 2y = 2 + 3\sin x \] and \[ y(0) = -\frac{7}{4}, \] then \( y\left( \frac{\pi}{6} \right) \) is:

• A. \( \frac{3}{4} \)
• B. \( \frac{4}{3} \)
• C. \( \frac{5}{2} \)
• D. \( -\frac{5}{2} \)

Hint:

When solving first-order linear differential equations, use the integrating factor method to simplify and solve for the function \( y \). Apply initial conditions to find the constants of integration.

Answer 1

The Correct Option is D

Step 1: Simplify the equation.
We are given the differential equation: \[ \sec x \frac{dy}{dx} - 2y = 2 + 3\sin x \] First, divide the entire equation by \( \sec x \) to simplify it: \[ \frac{dy}{dx} - 2y \sec x = 2 \sec x + 3 \sin x \sec x \]
Step 2: Solve the differential equation.
This is a first-order linear differential equation. The standard form of such an equation is: \[ \frac{dy}{dx} + P(x) y = Q(x) \] where \( P(x) \) and \( Q(x) \) are functions of \( x \). In this case, we can identify: \[ P(x) = -2 \sec x \quad \text{and} \quad Q(x) = 2 \sec x + 3 \sin x \sec x \] Now, solve the differential equation using the integrating factor method: \[ I(x) = e^{\int P(x) \, dx} = e^{\int -2 \sec x \, dx} \] The integral of \( -2 \sec x \) is \( -2 \ln |\sec x + \tan x| \), so the integrating factor is: \[ I(x) = |\sec x + \tan x|^{-2} \] Now multiply both sides of the original equation by the integrating factor to solve for \( y \).
Step 3: Apply the initial condition.
We are given the initial condition \( y(0) = -\frac{7}{4} \). Substitute \( x = 0 \) into the solution to find the constant of integration.
Step 4: Calculate \( y\left( \frac{\pi}{6} \right) \).
After solving the differential equation, substitute \( x = \frac{\pi}{6} \) into the solution to find: \[ y\left( \frac{\pi}{6} \right) = -\frac{5}{2} \] Thus, the correct answer is (4).