1.1.2. Example 2

Given: z=a²+x²+y²

In Example 2, we start with a function z represented by the equation z = a² + x² + y². Here, a is a constant and x, y are independent variables. This equation shows how z depends on both x and y, with a² as an arbitrary constant that needs to be eliminated to form a partial differential equation (PDE). The goal is to derive a PDE involving the partial derivatives associated with the independent variables.

Think of z as the height of a hill (z) at any point on a 2D plane represented by x (east-west) and y (north-south) coordinates. The constant a² represents a baseline height that is fixed, similar to how the sea level is constant while the hill's height varies.

Differentiate: ∂z/∂x = 2x, ∂z/∂y = 2y

The next step is to differentiate z partially with respect to x and y. This involves using the rules for differentiation. As a result, we find the first partial derivative of z with respect to x is 2x, and with respect to y, it's 2y. These derivatives (2x and 2y) help us understand how z changes as we change x and y, respectively.

Imagine you are walking on the hill mentioned earlier. As you go east (x-direction), the height increase (z) per step you take can be thought of as the derivative ∂z/∂x. The derivative tells you how steep the hill is at point x.

Let: p=2x, q=2y ⇒ x= p/2, y= q/2

In this step, we introduce new variables p and q to represent the derivatives we just calculated. We let p = 2x and q = 2y. This allows us to express x and y in terms of p and q, which simplifies our process of eliminating the constants. Thus, we write x as p/2 and y as q/2.

Continuing with the hill analogy, think of p and q as indicators of how much further you move in the east (x) and north (y) directions at a certain point on the hill. By organizing these movements into their own quantities (p and q), we simplify our calculations.

Substitute into original: z=a²+x²+y² = a² + (p/2)² + (q/2)² ⇒a² = z - (p² + q²)/4

Next, we substitute our expressions for x and y back into the original equation for z. This results in rewriting z as: z = a² + (p/2)² + (q/2)². By rearranging the equation, we isolate a², resulting in a² = z - (p² + q²)/4. This manipulation is crucial because it brings us closer to eliminating the arbitrary constant a².

Imagine we're trying to find out how high the hill (z) is without considering the fixed base height (a²). By adjusting p and q to account for how far you've stepped (thus changing x and y), we can express the hill's height in a more flexible way that doesn't depend on the fixed base.

Thus, the required PDE is: p² + q² = 4z

Finally, we eliminate the constant a², leading us to the partial differential equation p² + q² = 4z. This equation expresses the relationship between the derivatives (p and q) and the variable z without the arbitrary constant, thus forming a proper PDE for this system.

In our hill example, we now have a relationship that tells us how the steepness (p and q) relates to the height (z) over a portion of the hill. We've eliminated the constant baseline (a²), focusing solely on how shifts in terrain affect one another.

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