The given partial differential equation (PDE) is:

$p \cdot x + q \cdot y = z$

where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.

This equation can be solved using the method of characteristics. Let's proceed with the solution:

Step 1: Write down the characteristic equations

The characteristic equations for the PDE are:

$\frac{dx}{p} = \frac{dy}{q} = \frac{dz}{p \cdot x + q \cdot y}$

Step 2: Solve the characteristic equations

We solve these equations step by step:

1. First pair: $\frac{dx}{p} = \frac{dy}{q}$

   This implies: $\frac{dx}{x} = \frac{dy}{y}$ Integrating both sides, we get: $\ln x - \ln y = C_1 \quad \text{or} \quad \frac{x}{y} = C_1$ Thus, the first characteristic equation is: $x = C_1 y$

2. Second pair: $\frac{dz}{p \cdot x + q \cdot y} = \frac{dx}{p}$ Using the expression $p \cdot x + q \cdot y = z$, we have: $\frac{dz}{z} = \frac{dx}{x}$ Integrating both sides, we get: $\ln z = \ln x + C_2 \quad \text{or} \quad z = C_2 x$

Step 3: General solution

The general solution to the PDE is obtained by combining the solutions from the characteristic equations:

$z = F\left(\frac{x}{y}\right) \cdot x$

Where $F$ is an arbitrary function of the ratio $\frac{x}{y}$.

Therefore, the general solution to the PDE $px + qy = z$ is:

$z = x \cdot F\left(\frac{x}{y}\right)$

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you solve a PDE using the separation of variables method?
2. What are the characteristic curves in the method of characteristics?
3. Can this PDE have a particular solution instead of a general one?
4. How does the choice of the function $F$ affect the solution?
5. What are some applications of this type of PDE in physics or engineering?

Tip: When solving PDEs, identifying the appropriate method (like characteristics, separation of variables, or Fourier transform) is crucial for finding the solution efficiently.