To solve the partial differential equation $p + 3q = \tan(y - 3x)$, where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$, we can follow the method of characteristics.

Step 1: Write the PDE in the characteristic form

The equation $p + 3q = \tan(y - 3x)$ becomes: $\frac{\partial z}{\partial x} + 3\frac{\partial z}{\partial y} = \tan(y - 3x)$

This is a first-order linear partial differential equation of the form: $a(x, y) \frac{\partial z}{\partial x} + b(x, y) \frac{\partial z}{\partial y} = c(x, y)$ with $a(x, y) = 1$, $b(x, y) = 3$, and $c(x, y) = \tan(y - 3x)$.

Step 2: Set up the characteristic system

The characteristic system is given by: $\frac{dx}{1} = \frac{dy}{3} = \frac{dz}{\tan(y - 3x)}$

From the first two equations, we can solve for the characteristic curves: $\frac{dy}{dx} = 3 \quad \Rightarrow \quad y = 3x + C_1$ where $C_1$ is a constant along the characteristic curves.

Step 3: Use the characteristic equation for $z$

Now substitute $y = 3x + C_1$ into the original PDE: $\frac{dz}{dx} = \tan(y - 3x)$ Substitute $y - 3x = C_1$ into this equation: $\frac{dz}{dx} = \tan(C_1)$ This gives: $z = x \tan(C_1) + C_2$ where $C_2$ is another constant of integration.

Step 4: Express the solution in terms of $x$ and $y$

Since $C_1 = y - 3x$, the general solution can be written as: $z = x \tan(y - 3x) + f(y - 3x)$ where $f(y - 3x)$ is an arbitrary function of $y - 3x$, which comes from the general solution of the homogeneous part of the equation.

Final Solution:

The general solution to the PDE is: $z = x \tan(y - 3x) + f(y - 3x)$ where $f$ is an arbitrary function of $y - 3x$.

Would you like more details on any step, or have any questions?

Here are 5 related questions to consider:

1. How do we handle different types of first-order PDEs?
2. What is the method of characteristics in solving PDEs?
3. Can you solve nonlinear PDEs using the same method?
4. How can the arbitrary function $f(y - 3x)$ be determined in specific cases?
5. What other methods can be used to solve first-order linear PDEs?

Tip: In many PDE problems, recognizing the form of the equation and selecting an appropriate method (like characteristics) can significantly simplify the solution.