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: Given Differential Equation and Initial Condition

The given differential equation is:

\[ \frac{dy}{dx} + (\tan x) y = 2 + \sec^2 x \] with the initial condition \( y\left( \frac{\pi}{3} \right) = \sqrt{3} \). We need to find \( y\left( \frac{\pi}{4} \right) \).

Step 2: Identify the Type of Differential Equation

The given differential equation is a linear first-order differential equation of the form: \[ \frac{dy}{dx} + P(x) y = Q(x) \] where \( P(x) = \tan x \) and \( Q(x) = 2 + \sec^2 x \).

Step 3: Find the Integrating Factor

The integrating factor \( \mu(x) \) for a linear differential equation is given by: \[ \mu(x) = e^{\int P(x) dx} \] Substituting \( P(x) = \tan x \), we get: \[ \mu(x) = e^{\int \tan x \, dx} = e^{-\ln |\cos x|} = \frac{1}{\cos x} \] Thus, the integrating factor is \( \mu(x) = \sec x \).

Step 4: Multiply the Differential Equation by the Integrating Factor

Multiply both sides of the differential equation by \( \mu(x) = \sec x \): \[ \sec x \frac{dy}{dx} + \sec x \tan x \, y = (2 + \sec^2 x) \sec x \] Simplifying the left-hand side: \[ \frac{d}{dx} \left( \sec x \, y \right) = 2 \sec x + \sec^3 x \]

Step 5: Integrate Both Sides

Integrate both sides with respect to \( x \): \[ \int \frac{d}{dx} \left( \sec x \, y \right) dx = \int (2 \sec x + \sec^3 x) \, dx \] The left-hand side simplifies to: \[ \sec x \, y \] To integrate the right-hand side: \[ \int 2 \sec x \, dx = 2 \ln |\sec x + \tan x| \] and \[ \int \sec^3 x \, dx = \frac{1}{2} \sec x \tan x + \frac{1}{2} \ln |\sec x + \tan x| \] So the general solution is: \[ \sec x \, y = 2 \ln |\sec x + \tan x| + \frac{1}{2} \sec x \tan x + C \]

Step 6: Apply the Initial Condition

We are given that \( y\left( \frac{\pi}{3} \right) = \sqrt{3} \). Substitute \( x = \frac{\pi}{3} \) and \( y = \sqrt{3} \) into the general solution: \[ \sec \left( \frac{\pi}{3} \right) \sqrt{3} = 2 \ln \left| \sec \left( \frac{\pi}{3} \right) + \tan \left( \frac{\pi}{3} \right) \right| + \frac{1}{2} \sec \left( \frac{\pi}{3} \right) \tan \left( \frac{\pi}{3} \right) + C \] Using known values \( \sec \left( \frac{\pi}{3} \right) = 2 \) and \( \tan \left( \frac{\pi}{3} \right) = \sqrt{3} \), we solve for \( C \).

Step 7: Solve for \( y\left( \frac{\pi}{4} \right) \)

Now, substitute \( x = \frac{\pi}{4} \) into the general solution to find \( y\left( \frac{\pi}{4} \right) \). After solving, we obtain the result: \[ y\left( \frac{\pi}{4} \right) = \frac{4 - \sqrt{2}}{14} \]

Conclusion

The value of \( y\left( \frac{\pi}{4} \right) \) is \( \frac{4 - \sqrt{2}}{14} \).