In the context of partial differential equations, the symbols \( p \), \( q \), \( r \), and \( s \) typically represent the following derivatives:

- \( p = \frac{\partial z}{\partial x} \): The partial derivative of \( z \) with respect to \( x \).
- \( q = \frac{\partial z}{\partial y} \): The partial derivative of \( z \) with respect to \( y \).
- \( r = \frac{\partial^2 z}{\partial x^2} \): The second partial derivative of \( z \) with respect to \( x \).
- \( s = \frac{\partial^2 z}{\partial y^2} \): The second partial derivative of \( z \) with respect to \( y \).

Additionally, you might encounter:
- \( t = \frac{\partial^2 z}{\partial x \partial y} \): The mixed partial derivative of \( z \) with respect to both \( x \) and \( y \).

Given this notation, the PDE derived from the expression \( z = x f(x+y) + g(x+y) \) involves these symbols:

### Step 1: Express derivatives using \( p \), \( q \), \( r \), \( s \)
1. \( p = \frac{\partial z}{\partial x} = f(x+y) + x f'(x+y) + g'(x+y) \)
2. \( q = \frac{\partial z}{\partial y} = x f'(x+y) + g'(x+y) \)

### Step 2: Eliminate the arbitrary functions
Subtract \( q \) from \( p \):
\[
p - q = f(x+y)
\]
Differentiate with respect to \( y \):
\[
\frac{\partial p}{\partial y} - \frac{\partial q}{\partial y} = f'(x+y)
\]
This can also be written as:
\[
\frac{\partial^2 z}{\partial y \partial x} - \frac{\partial^2 z}{\partial y^2} = f'(x+y)
\]
Or using the notation:
\[
t - s = f'(x+y)
\]

Finally, eliminating the function \( g(x+y) \) and combining these gives the final PDE.

Would you like further details on how to obtain the complete PDE?

---

**Relative questions:**

1. What would the mixed partial derivative \( \frac{\partial^2 z}{\partial x \partial y} \) represent in this context?
2. How do these symbols \( p \), \( q \), \( r \), and \( s \) relate to the classification of PDEs?
3. Can you provide another example where you use the \( p \), \( q \), \( r \), and \( s \) notation in a PDE?
4. How would you interpret the physical meaning of these derivatives in a real-world application?
5. How does this notation help simplify the process of deriving PDEs?

**Tip:** Using standardized symbols like \( p \), \( q \), \( r \), and \( s \) in PDEs helps streamline the process of derivation and makes it easier to recognize patterns and solutions.

Explore the meanings and applications of symbols p, q, r, and s in partial differential equations (PDEs). Learn how these symbols represent various derivatives of the function z, aiding in the derivation of PDEs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation