Let's break down the problems shown in the image one by one:

Q1:

(a) The following terms:

1. Arbitrary constant: In differential equations, arbitrary constants represent general solutions to the equation before specific boundary conditions or initial conditions are applied.
2. Boundary value problems: These are differential equations with conditions specified at the boundaries of the domain where the solution is defined. Typical examples are physical systems constrained at specific points.
3. Nonlinear operator: A mathematical operator that doesn't satisfy the linearity property (i.e., additivity and homogeneity). Nonlinear operators often lead to complex, non-superimposable solutions.

(2) Classify the equation:

The equation $\frac{\partial^2 u}{\partial t^2} = xyt$ is a second-order partial differential equation (PDE). Since the highest derivative with respect to time is second order, and there are no terms like $u$ itself or its first derivative in the equation, it is linear with respect to $u$ but nonlinear overall due to the multiplication of $xyt$.

(b) Show linearity of the Laplace operator:

You need to demonstrate the linearity property of the Laplace transform $\mathcal{L}$. If you take two functions $f_1$ and $f_2$ with constants $c_1$ and $c_2$, the operator should satisfy: $\mathcal{L}\{c_1 f_1 + c_2 f_2\} = c_1 \mathcal{L}\{f_1\} + c_2 \mathcal{L}\{f_2\}$ This is a basic property of linear operators.

Q2:

(a) Solve the differential equation:

The equation is: $x^2 y''' + y' - \frac{y}{x} = 3x^3$ This is a third-order linear differential equation with variable coefficients. It can be solved using standard methods such as reduction of order or applying specific methods like variation of parameters.

(b) Solve the differential equation:

$y^{(4)} - 9y = 0$ This is a fourth-order homogeneous linear differential equation. The characteristic equation is: $r^4 - 9 = 0 \quad \Rightarrow \quad r = \pm 3$ Hence, the general solution is: $y(x) = c_1 e^{3x} + c_2 e^{-3x} + c_3 x e^{3x} + c_4 x e^{-3x}$ Using the initial conditions $y(0) = 1, y'(0) = 0, y''(0) = -3, y'''(0) = 0$, the constants $c_1, c_2, c_3, c_4$ can be determined.

Q3:

(a) Solve the equation:

$z = \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2$ This is a nonlinear partial differential equation. The method of characteristics or separation of variables may be useful for solving this type of equation, depending on the boundary conditions or assumptions about the variables.

(b) Laplace transform:

You need to solve the PDE: $\frac{\partial u}{\partial t} = 3 \frac{\partial u}{\partial x}$ with the initial condition $u(x, 0) = 4e^{-2x}$. Using the Laplace transform in time and then applying inverse transforms to solve the equation is the approach for this problem.

Do you want detailed solutions for any of these problems?

Here are 5 follow-up questions to explore further:

1. How do you classify partial differential equations (PDEs) based on order and linearity?
2. What are the general methods to solve linear differential equations with constant coefficients?
3. What is the method of separation of variables, and how is it applied in solving PDEs?
4. Can you provide a detailed solution for a boundary value problem using Laplace transforms?
5. How does the method of characteristics work for solving nonlinear PDEs?

Tip: For solving PDEs, always check if boundary or initial conditions can simplify the equation early on.