Question

If for the solution curve \( y = f(x) \) of the differential equation \[ \frac{dy}{dx} + (\tan x) y = 2 + \sec^2 x, \quad y(\frac{\pi}{3}) = \sqrt{3}, \] \(\text{then}\) \( y(\frac{\pi}{4}) \) is equal to:

• A. \( \frac{3 + \sqrt{3}}{2} \)
• B. \( \frac{3 + 1}{(1 + \sqrt{3})} \)
• C. \( \frac{3 + \sqrt{3}}{(4 + \sqrt{3})} \)
• D. \( \frac{4 - \sqrt{2}}{14} \)

Hint:

In solving such differential equations, identify the integrating factor carefully and use it for efficient integration of the equation.

Answer 1

The Correct Option is D

Step 1: The given differential equation is of the form: \[ \frac{dy}{dx} + (\tan x) y = 2 + \sec^2 x \] \(\text{To solve the above, the integrating factor is:}\) \[ I = \sec x \] \(\text{Multiplying through by the integrating factor:}\) \[ \sec x \frac{dy}{dx} + (\tan x) \sec x y = 2 + \sec^2 x \] \(\text{Simplifying and integrating both sides:}\) \[ \frac{d}{dx} \left( y \cdot \sec x \right) = 2 \cdot \sec x \] \(\text{Integrating both sides:}\) \[ y \cdot \sec x = 2 \ln \left( \sec x + \tan x \right) + C \] \(\text{Using the given initial condition:}\) \[ y \cdot \sec \left( \frac{\pi}{3} \right) = 2 \ln \left( \sec \left( \frac{\pi}{3} \right) + \tan \left( \frac{\pi}{3} \right) \right) + C \] \[ \sqrt{3} = 2 \cdot \ln \left( \sqrt{3} + \frac{\sqrt{3}}{2} \right) + C \] \(\text{From the calculations, we obtain the value of C.}\) \(\text{Now substitute and find the value of \( y(\frac{\pi}{4}) \) using the value of C.}\)