Eliminating f to Form a Partial Differential Equation: A Step-by-Step Guide

Introduction

Elimination of a function in the context of partial differential equations (PDEs) is a powerful technique used to transform complex relations into simpler, more manageable forms. This article provides a detailed step-by-step guide on how to eliminate a function f (from z f(x - y)) to obtain a partial differential equation (PDE). The process involves differentiation with respect to variables and the application of standard PDE notations. By the end of this guide, you will have a thorough understanding of the underlying mathematical principles and practical steps to form a PDE using the elimination technique.

Unraveling the Problem

Consider the given relation:

z f(x - y).

To derive a partial differential equation by eliminating the function f, we need to differentiate both sides of the equation with respect to the variables x and y. This will allow us to systematically eliminate the function f and arrive at a relationship involving the partial derivatives of z.

Step 1: Differentiation with Respect to x

The first step is to differentiate the given relation with respect to x:

z_x ?/?x[f(x - y)]

Using the chain rule, we have:

z_x f'(x - y) * ?(x - y)/?x

Since the derivative of x - y with respect to x is 1, we get:

z_x f'(x - y) * 1 f'(x - y)

Let's denote the first partial derivative as p, so:

z_x p f'(x - y)

Step 2: Differentiation with Respect to y

Next, we differentiate the given relation with respect to y:

z_y ?/?y[f(x - y)]

Again, using the chain rule, we have:

z_y f'(x - y) * ?(x - y)/?y

Since the derivative of x - y with respect to y is -1, we get:

z_y f'(x - y) * (-1) -f'(x - y)

Let's denote the second partial derivative as q, so:

z_y q -f'(x - y)

Step 3: Eliminating f

To eliminate the function f, we now have:

z_x p f'(x - y)

z_y q -f'(x - y)

By comparing these equations, we can see that:

p -q

In standard PDE notations, p and q are often written as P and Q. Therefore, the elimination of f gives us:

PQ 0

This is the required partial differential equation.

Conclusion

In summary, the process of eliminating f from z f(x - y) involves differentiating both sides of the equation with respect to x and y, and then using the chain rule to find the first-order partial derivatives. By setting the resulting derivatives in relation to each other, we can eliminate the function f and arrive at a partial differential equation. The final PDE obtained is PQ 0, where P and Q represent the partial derivatives of z with respect to x and y, respectively.

Related Keywords

Partial Differential Equation Eliminating f Mathematical Derivation

References

For a deeper understanding of partial differential equations and the methods to solve them, consult the following resources: Boyce, W. E., DiPrima, R. C. (2001). Elementary differential equations. John Wiley Sons. Evans, L. C. (1998). Partial differential equations. American Mathematical Society.