The problem in the image deals with solving a second-order differential equation using the method of variation of parameters. The equation given is:

$y'' + y = \sec^2(t)$

The homogeneous solution $y_h$ is provided as:

$y_h = C_1 \cos(t) + C_2 \sin(t)$

Step-by-Step Solution

1. Solve the homogeneous equation: The homogeneous differential equation corresponding to the problem is: $y'' + y = 0$ The characteristic equation for this is: $r^2 + 1 = 0$ Solving for $r$: $r = \pm i$ This gives the general solution of the homogeneous equation: $y_h = C_1 \cos(t) + C_2 \sin(t)$ which is already provided.

2. Variation of Parameters: To solve the non-homogeneous equation using variation of parameters, we assume a particular solution of the form: $y_p = u_1(t) \cos(t) + u_2(t) \sin(t)$ where $u_1(t)$ and $u_2(t)$ are functions to be determined.

3. Determine $u_1(t)$ and $u_2(t)$: We need to solve for $u_1(t)$ and $u_2(t)$ using the system of equations derived from: $u_1'(t) \cos(t) + u_2'(t) \sin(t) = 0$ and $-u_1'(t) \sin(t) + u_2'(t) \cos(t) = \sec^2(t)$

   Solving these equations, we obtain: $u_1'(t) = -\sec^2(t) \sin(t)$ $u_2'(t) = \sec^2(t) \cos(t)$

4. Integrate to find $u_1(t)$ and $u_2(t)$:

   • Integrating $u_1'(t)$: $u_1(t) = -\int \sec^2(t) \sin(t) \, dt$ This integral can be solved using substitution.

   • Integrating $u_2'(t)$: $u_2(t) = \int \sec^2(t) \cos(t) \, dt$ Similarly, solve this integral.

5. Construct the particular solution: After finding $u_1(t)$ and $u_2(t)$, substitute them back into the particular solution $y_p = u_1(t) \cos(t) + u_2(t) \sin(t)$.

6. General solution: The general solution to the differential equation is: $y(t) = y_h + y_p = C_1 \cos(t) + C_2 \sin(t) + y_p$

Would you like more detailed steps on the integration process or any specific part of the solution?

Follow-up Questions:

1. How do we solve the integrals in the variation of parameters method?
2. What is the significance of the homogeneous solution in solving differential equations?
3. Can variation of parameters be used for any non-homogeneous differential equation?
4. How do we verify that the solution satisfies the original differential equation?
5. What are the alternative methods for solving second-order differential equations?

Tip:

When solving differential equations, always check if the particular solution you found is valid by substituting it back into the original equation.