1.2. Formation of PDEs by Eliminating Arbitrary Functions

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

The formation of Partial Differential Equations (PDEs) by eliminating arbitrary functions begins with identifying a function that includes these arbitrary functions. This method involves the differentiation of the function with respect to its variables and then using relationships between the derivatives to eliminate the arbitrary functions. This process results in obtaining a PDE that captures the behavior of the system being analyzed.

Think of eliminating arbitrary functions like uncovering the truth behind the effects of different variables in a recipe. If you have an ingredient that varies (like spices), you first analyze how the flavor changes with different amounts and then simplify the recipe to express it in terms of fixed quantities rather than those variable spices.

📌 Example 3: Given: z=f(x²+y²) Let: u=x²+y², so z=f(ν) Differentiate: ∂z∂u=f′(u)⋅=f′(u)⋅2x⇒p=2xf′(u) ∂x ∂x ∂z df′(u)⋅2y⇒q=2yf′(u) ∂y Now, eliminate f′(u): Divide:

p/q = f′(u), = f′(u)⇒ ⇒ 2x 2y Thus, This is the required PDE.

In Example 3, we start with a function z that depends on an arbitrary function f of the sum of squares of x and y. By introducing a new variable ν for x² + y², we differentiate z to find its relationships with respect to x and y. We introduce new variables p and q to simplify the equation and eliminate f' (the derivative of the arbitrary function). The relationships obtained through differentiation lead us to form a PDE, which describes how z changes based on x and y without explicitly needing the function f.

Imagine you are tracking the growth of a plant based on different conditions (rather than just its height). By measuring how temperature (x) and sunlight (y) affect the plant, you create a graph (z) to represent the growth. Instead of detailing every single plant's unique growth path (like the arbitrary function f), you simplify the observation into general trends (the PDE) that show how environmental factors relate to the growth.

📌 Example 4: Given: z=f(x+y)+g(x−y) Let: u=x+y, v=x−y⇒z=f(ν)+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 influenced by two arbitrary functions: f and g. By introducing new variables to represent combinations of x and y, we can differentiate z and establish relationships between these functions. By adding and subtracting the equations derived from differentiation, we can express the individual derivatives of f and g in terms of new variables p and q. This approach allows us to eliminate the arbitrary functions and create a more general PDE that describes the behavior of z based on x and y.

Consider a complex music composition that involves several instruments (like f and g). The overall melody (z) is created by mixing different sounds from individual instruments. By analyzing how each instrument contributes to the overall sound, instead of depicting each instrument separately, you simplify into the chords formed (the PDE). The final composition correlates the sounds flexibly yet captures the essence required for the tune.

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