The image you've uploaded contains a list of mathematical problems, mostly focused on solving differential equations. Here’s a transcription and brief overview of the questions I can extract:

PART-A

1. Solve $(D + 1)y = 0$.
2. Find the P.I. of $(D^2 + 2) = \sin 2x$.
3. Find the P.I. of $(D^2 + 4) = \cos 2x$.
4. Form a PDE by eliminating the arbitrary constants from $z = f(xa) + g(y + bt)$.
5. Solve $(D - 1)(D + 1)y = \log x$.

Module-1: PART-B 6. Solve $\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = \cos 3x$. 7. Solve $3 \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 2y = 10(x + 1)$. 8. Solve $(D - 1)(D + 1)y = \log x$.

Module-2 9. Solve $\frac{\partial^2 z}{\partial x^2} = \cos(2x + 3y)$. 10. Solve $\frac{\partial^2z}{\partial x^2} + z = 0$, given that $z = e^y$ and $\frac{\partial z}{\partial x} = 1$ when $x = 0$. 11. Solve $\sqrt{p - z} | p + (z - x) q | = (x - y)$.

Would you like a solution for one of these questions? Let me know if you need detailed explanations for any specific problem or further clarifications!

Follow-up questions:

1. How do you find the complementary function (C.F.) for second-order linear differential equations?
2. What are the steps to eliminate arbitrary constants from a function in PDEs?
3. Can you explain the method to find the particular integral (P.I.) of a non-homogeneous differential equation?
4. How can you solve partial differential equations involving multiple variables?
5. What methods can be used to solve homogeneous versus non-homogeneous differential equations?

Tip:

When solving differential equations, always check if the equation is homogeneous or non-homogeneous before proceeding to find the complementary function and particular integral.