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.

Explore the step-by-step solution to the linear PDE p + 3q = tan(y - 3x) using the method of characteristics. Learn about first-order PDEs, the characteristic system, and how to find the general solution. Suitable for advanced undergraduate or graduate students.

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