I see that you've uploaded an image with two mathematical questions. Let's break down each question and provide the solution:

Question 1

It looks like this is a multiple-choice question regarding the value of something (likely a constant or result). However, without seeing the actual content in the image, I can't fully analyze the question. Could you provide more details or clarify the mathematical context for question 1?

Question 2

The question asks about the general solution to a second-order linear differential equation $y'' + p_1y' + p_2y = 0$, with $y_1$ and $y_2$ being two linearly independent solutions. The problem appears to be about linear combinations of these solutions.

• The options present combinations of $y_1$ and $y_2$, with constant coefficients $C_1$ and $C_2$, and the task is to determine which expression represents a general solution.

Analysis:

• For a linear second-order homogeneous differential equation like this, the general solution is given by a linear combination of two linearly independent solutions $y_1$ and $y_2$.
• The general form is: $y(x) = C_1 y_1(x) + C_2 y_2(x)$
• This linear combination is indeed the general solution of the differential equation, so option A would be correct.

Solution:

The correct answer to Question 2 is A: $C_1 y_1 + C_2 y_2$ is the general solution to the differential equation.

Would you like to dive into any further details on these questions? Or do you have any specific questions about the concepts? Here are some related topics to explore:

1. How to solve second-order linear differential equations.
2. Understanding the concept of linearly independent solutions in differential equations.
3. How the characteristic equation is formed for such differential equations.
4. Explanation of initial conditions and how they affect the solution.
5. The role of constants $C_1$ and $C_2$ in the general solution.

Tip: When solving second-order linear differential equations, always check if the given solutions are linearly independent, as this is key to forming the correct general solution.