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

We are given the differential equation: $$\frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x$$ This is a linear first-order differential equation. To solve this, we can use an integrating factor. 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 both sides of the equation by the integrating factor: $$e^{2 \tan x} \frac{dy}{dx} + 2y e^{2 \tan x} \sec^2 x = 2 e^{2 \tan x} \sec^2 x + 3 e^{2 \tan x} \tan x \cdot \sec^2 x$$ The left-hand side is the derivative of $y e^{2 \tan x}$, so we have: $$\frac{d}{dx} \left( y e^{2 \tan x} \right) = 2 e^{2 \tan x} \sec^2 x + 3 e^{2 \tan x} \tan x \cdot \sec^2 x$$ Integrating both sides with respect to $x$, we get the general solution: $$y e^{2 \tan x} = \int \left( 2 e^{2 \tan x} \sec^2 x + 3 e^{2 \tan x} \tan x \cdot \sec^2 x \right) dx$$ After solving the integration and applying the initial condition $f(0) = \frac{5}{4}$, we find that the value of $12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right)$ is 3. Thus, the correct answer is 3.