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The given problem involves finding the value of \( B \) in a particular solution to a second-order linear differential equation using variation of parameters.

We are given the differential operator form of the equation:

\[
(D^2 - 2D + 1)y = e^t \sec t \tan t
\]

We are also provided with a particular solution of the form:

\[
y_p = A e^t + B t e^t
\]

Here, \( D \) represents differentiation with respect to \( t \).

### Steps to Solve:

1. **Homogeneous Solution**: The operator \( D^2 - 2D + 1 \) suggests the homogeneous equation is:
   \[
   (D^2 - 2D + 1) y_h = 0
   \]
   The characteristic equation for this differential operator is:
   \[
   r^2 - 2r + 1 = 0
   \]
   Solving this, we get a repeated root \( r = 1 \). Therefore, the general solution to the homogeneous equation is:
   \[
   y_h = (C_1 + C_2 t) e^t
   \]
   
2. **Particular Solution Using Variation of Parameters**: To solve for the particular solution, we apply variation of parameters. We assume a particular solution of the form:
   \[
   y_p = A(t) e^t + B(t) t e^t
   \]
   Substituting this form into the non-homogeneous equation \( (D^2 - 2D + 1)y = e^t \sec t \tan t \), we can solve for the functions \( A(t) \) and \( B(t) \).

3. **Coefficient Comparison**: Given that the non-homogeneous term is \( e^t \sec t \tan t \), and the particular solution has the form \( y_p = A e^t + B t e^t \), we can compare terms to isolate \( B \).

Based on the setup, the correct answer for \( B \) should match one of the choices. Using variation of parameters typically results in expressions involving functions of \( t \), and considering the form of the non-homogeneous term, the correct answer is **\( B = \sec t \)**.

Thus, the correct option is **B. \( \sec t \)**.

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Do you want a more detailed explanation or have any questions? Here are 5 relative questions to consider:

1. How do you apply the method of variation of parameters for second-order differential equations?
2. What are the steps to solving a characteristic equation for homogeneous differential equations?
3. How do you determine particular solutions for non-homogeneous differential equations?
4. What is the significance of repeated roots in the characteristic equation of a differential operator?
5. How does the form of the non-homogeneous term affect the particular solution?

**Tip**: When solving differential equations, always check if the roots of the characteristic equation are distinct, repeated, or complex. This affects the structure of the homogeneous solution.

If the following differential equation has a particular solution expressed as y_p = A e^t + B t e^t, find B using variation of parameters: (D^2 - 2D + 1)y = e^t sec t tan t.

Find the value of B in the particular solution y_p = A e^t + B t e^t of the second-order differential equation (D^2 - 2D + 1)y = e^t sec t tan t using the method of variation of parameters. This problem demonstrates key concepts in differential equations and is suitable for undergraduate-level students.

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