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: Initial Problem Statement

The problem involves a first-order linear differential equation and an initial condition. The equation is:

\[ \frac{dy}{dx} + (\tan x) y = 2 + \sec^2 x \]

The initial condition is \( y\left( \frac{\pi}{3} \right) = \sqrt{3} \). The objective is to determine the value of \( y\left( \frac{\pi}{4} \right) \).

Step 2: Classification of the Differential Equation

The given differential equation is identified as a linear first-order differential equation, conforming to the standard form:

\[ \frac{dy}{dx} + P(x) y = Q(x) \]

In this specific case, \( P(x) = \tan x \) and \( Q(x) = 2 + \sec^2 x \).

Step 3: Calculation of the Integrating Factor

The integrating factor \( \mu(x) \) for a linear differential equation is computed using the formula:

\[ \mu(x) = e^{\int P(x) dx} \]

Substituting \( P(x) = \tan x \), the integrating factor is derived as:

\[ \mu(x) = e^{\int \tan x \, dx} = e^{-\ln |\cos x|} = \frac{1}{\cos x} \]

Therefore, the integrating factor is \( \mu(x) = \sec x \).

Step 4: Multiplication by the Integrating Factor

Both sides of the differential equation are multiplied by the integrating factor \( \mu(x) = \sec x \):

\[ \sec x \frac{dy}{dx} + \sec x \tan x \, y = (2 + \sec^2 x) \sec x \]

The left-hand side is then simplified to the derivative of the product of the integrating factor and \( y \):

\[ \frac{d}{dx} \left( \sec x \, y \right) = 2 \sec x + \sec^3 x \]

Step 5: Integration of Both Sides

Integration is performed on both sides of the equation 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 \( \sec x \, y \). The integrals on the right-hand side are evaluated as:

\[ \int 2 \sec x \, dx = 2 \ln |\sec x + \tan x| \]

and

\[ \int \sec^3 x \, dx = \frac{1}{2} \sec x \tan x + \frac{1}{2} \ln |\sec x + \tan x| \]

The general solution is thus obtained:

\[ \sec x \, y = 2 \ln |\sec x + \tan x| + \frac{1}{2} \sec x \tan x + C \]

Step 6: Application of the Initial Condition

The initial condition \( y\left( \frac{\pi}{3} \right) = \sqrt{3} \) is used to find the constant of integration \( C \). Substituting \( x = \frac{\pi}{3} \) and \( y = \sqrt{3} \) into the general solution yields:

\[ \sec \left( \frac{\pi}{3} \right) \sqrt{3} = 2 \ln \left| \sec \left( \frac{\pi}{3} \right) + \tan \left( \frac{\pi}{3} \right) \right| + \frac{1}{2} \sec \left( \frac{\pi}{3} \right) \tan \left( \frac{\pi}{3} \right) + C \]

Using the known values \( \sec \left( \frac{\pi}{3} \right) = 2 \) and \( \tan \left( \frac{\pi}{3} \right) = \sqrt{3} \), the value of \( C \) is determined.

Step 7: Determination of \( y\left( \frac{\pi}{4} \right) \)

With the constant \( C \) determined, \( x = \frac{\pi}{4} \) is substituted into the general solution to find \( y\left( \frac{\pi}{4} \right) \). The calculated value is:

\[ y\left( \frac{\pi}{4} \right) = \frac{4 - \sqrt{2}}{14} \]

Conclusion

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