Here is the analysis and solution for the given questions in the image:

Question 1:

The elimination of arbitrary constants from the complete solution of a PDE gives the:

• Correct Answer: (c) Both a and b
  • Explanation:
    • Eliminating arbitrary constants from the general solution of a PDE results in a singular solution.
    • The general solution encompasses all possible solutions, while the singular solution is derived under specific conditions.

Question 2:

The PDE $y - x = p + q$ is of the standard type:

• Correct Answer: (a) Type I
  • Explanation:
    • $y - x = p + q$ fits the format of a first-order PDE of standard type. Standard types are classified based on their forms, and this equation corresponds to Type I classification.

Question 3:

Parseval's theorem in Fourier series relates:

• Correct Answer: (b) R.M.S value with Fourier coefficients of a function
  • Explanation:
    • Parseval’s theorem states that the total energy (or root mean square, R.M.S) of a function is equal to the sum of the squares of its Fourier coefficients. It is a key result in Fourier analysis for energy conservation.

If you need further clarification on any question or topic, feel free to ask!

Follow-Up Questions:

1. How can we derive a singular solution from a PDE's general solution step-by-step?
2. What are the different types of first-order PDEs, and how are they classified?
3. Can you explain Parseval’s theorem in a detailed mathematical derivation?
4. What are some real-world applications of Parseval’s theorem in signal processing?
5. How does the classification of PDEs (e.g., Type I-IV) affect their methods of solution?

Tip:

When solving PDEs, always verify whether the solution satisfies both the equation and any given boundary or initial conditions.