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.

Learn how to solve the partial differential equation (PDE) px+qy=z using the method of characteristics. This solution involves identifying characteristic equations, solving them step-by-step, and deriving the general solution for the PDE. Suitable for advanced students and researchers.

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