1.2.1. Method

📘 Method: If a function involves one or more arbitrary constants, we differentiate partially with respect to the independent variables and eliminate the constants.

In this method, when we have an equation that includes arbitrary constants, the first step is to take partial derivatives of the function with respect to its independent variables. This helps us to identify and isolate the arbitrary constants. After differentiating, we set up relationships to express the derived variables (let's call them p and q) in terms of these constants. Following this, we substitute back into the original equation and try to eliminate the constant, eventually leading us to a partial differential equation (PDE).

Think of it like a jigsaw puzzle where some pieces are missing (the arbitrary constants). By fitting the pieces (differentiating), you can find out which ones belong to the final image (the PDE). As you fit more pieces in, you start to see the complete picture emerge.

📌 Example 1: Given: z=ax+b y+c where a,b,c are arbitrary constants. Step 1: Differentiate partially with respect to x and y: ∂z ∂z a, =b ∂x ∂ y Let: ∂z ∂z p= ,q= ∂x ∂ y So: p=a,q=b Substitute into original equation: z=px+q y+c⇒c=z−px−qy Now eliminate c: No constants left to eliminate further. Hence, the PDE is: ∂2z z=px+q y+(z−px−qy)⇒This equation is too trivial; typically, we express it as: =0 ∂x∂ y

In Example 1, we start with a function z that contains the arbitrary constants a, b, and c. The first step is to differentiate z with respect to the variables x and y, resulting in derivatives that are equal to those constants. We then define p and q to simplify our notation. By substituting p and q back into the original equation, we express c in terms of z, leading to the removal of arbitrary constants. Ultimately, we derive a simple PDE, which is identified as trivial since it leads us to express it as equal to zero.

Imagine you're cooking and have a recipe that includes some 'to taste' ingredients (the constants). By tasting and adjusting (differentiating), you can figure out how much of each ingredient you actually need to create a dish that satisfies your taste (the PDE). Once you find the right amounts, you realize that some ingredients aren't needed at all (the trivial PDE).

📌 Example 2: Given: z=a2 +x2 +y2 Differentiate: ∂z ∂z =2x, =2 y ∂x ∂ y Let: p q p=2x,q=2y⇒x= , y= 2 2 Substitute into original: z=a2 +x2 +y2 =a2 +( p)2 +(q)2 ⇒a2 =z− p2 +q2 2 2 4 Thus, the required PDE is: p2 +q2 4 z=p2 +q2 +4a2⇒Eliminatea2:4z=p2 +q2 +4(z− )⇒Simplify to form the PDE 4

In Example 2, we begin with a function z that includes arbitrary constants. Differentiating with respect to x and y gives us new variables p and q. After this, we substitute these variables back into the original equation. We aim to express the constants in terms of the quantities we have derived. By adjusting and rearranging the terms carefully, we eventually eliminate the constants to arrive at a second-order PDE representing the relationship between the variables of the system.

Consider building a model of a structure where some elements are adjustable (the constants). By modifying the size and shape of parts (differentiating), you can determine the actual dimensions needed that will fit together perfectly (the PDE). Once you have this model, you can remove the flexible parts to get a solid understanding of the structure's final design.

📘 Method: If the function involves arbitrary functions, we differentiate and eliminate the function(s) using the given relationships.

When working with functions that involve arbitrary functions instead of constants, we follow a similar approach. First, we take partial derivatives and establish relationships between the variables. Once we have these derivatives expressed in terms of the arbitrary functions, we can eliminate those functions using the relationships identified. This process helps us arrive at a PDE that reflects the dependent relationships of the variables involved.

Imagine you’re solving a mystery with clues (the arbitrary functions). Each time you collect a clue (take a derivative), you establish how the clues relate to each other. By connecting the dots and removing the irrelevant hints (eliminating the functions), you unveil the actual story behind the mystery (arriving at the PDE).

📌 Example 3: Given: z=f(x2 +y2) Let: u=x2 +y2,soz=f(u) Differentiate: ∂z ∂u =f′ (u)⋅ =f′ (u)⋅2x⇒p=2xf′ (u) ∂x ∂x ∂z =f′ (u)⋅2y⇒q=2yf′ (u) ∂ y Now, eliminate f′ (u): Divide: p q p q =f′ (u), =f′ (u)⇒ = ⇒p y=qx 2x 2 y 2x 2 y This is the required PDE.

Example 3 takes a function defined by an arbitrary function f that depends on a combination of variables. After differentiating, we define new variables p and q based on those derivatives. The goal is to eliminate the function f itself. We can achieve this by relating p and q through the operations of division to express the arbitrary function in a different form. This leads us to derive a PDE that captures the relationship of the original function with respect to x and y.

Think of this as trying to decode a secret language (the arbitrary function). You gather different clues (the derivatives) that help reveal how words relate to each other. By carefully piecing together these clues and recognizing the patterns, you can uncover the true message or meaning behind the language (the PDE).

📌 Example 4: Given: z=f(x+y)+g(x− y) Let: u=x+y,v=x− y⇒z=f(u)+g(v) Differentiate: ∂z ∂z =f′ (u)+g′ (v), =f′ (u)−g′ (v) ∂x ∂ y Add and subtract: p=f′ (u)+g′ (v),q=f′ (u)−g′ (v) Add: p+q p+q=2f′ (u)⇒f′ (u)= 2 Subtract: p−q p−q=2g′ (v)⇒g′ (v)= 2 Now differentiate again and eliminate f and g to form a second-order PDE.

In Example 4, we have a function z that is defined by two arbitrary functions f and g. We start by differentiating the function with respect to x and y. From this point, we can express the derivatives in terms of new variables p and q. By adding and subtracting these equations, we isolate the derivatives of f and g. This allows us to eliminate both f and g entirely through further differentiation, culminating in the creation of a second-order PDE that connects the relationships among the combined functions.

Imagine you’re blending two recipes (the arbitrary functions). As you combine them, each recipe gives you a hint about the flavors (the derivatives). By tasting and adjusting the blend (differentiating), you can identify the dominant flavors (isolate f and g) and create a completely new dish (the PDE) that represents the best of both original recipes.

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