Question

If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and

and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When solving first-order linear differential equations, always use the method of integrating factors. This simplifies the problem and helps find the general solution.

Answer 1

The Correct Option is C

The given differential equation is: \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] This is a first-order linear differential equation. An integrating factor is used for its solution. The equation can be rewritten as: \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] The integrating factor is \( e^{\int 2 \sec^2 x dx} = e^{2 \tan x} \). Multiplying the equation by the integrating factor yields: \[ e^{2 \tan x} \frac{dy}{dx} + 2y e^{2 \tan x} \sec^2 x = 2 e^{2 \tan x} \sec^2 x + 3 e^{2 \tan x} \tan x \cdot \sec^2 x \] The left-hand side is equivalent to the derivative of \( y e^{2 \tan x} \): \[ \frac{d}{dx} \left( y e^{2 \tan x} \right) = 2 e^{2 \tan x} \sec^2 x + 3 e^{2 \tan x} \tan x \cdot \sec^2 x \] Integrating both sides with respect to \( x \) provides the general solution: \[ y e^{2 \tan x} = \int \left( 2 e^{2 \tan x} \sec^2 x + 3 e^{2 \tan x} \tan x \cdot \sec^2 x \right) dx \] Following integration and application of the initial condition \( f(0) = \frac{5}{4} \), the value of \( 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \) is determined to be 3. Therefore, the correct answer is 3.