1.1.1. Method

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

To form a Partial Differential Equation (PDE) from a function that has arbitrary constants, we begin by differentiating that function with respect to its independent variables. By taking these derivatives, we can represent the constants in terms of the derivatives. Once we have expressed the variables in terms of their derivatives, we substitute back into the original function to derive the PDE. This method essentially allows us to eliminate the arbitrary constants from our equations.

Imagine you have a recipe for a cake that includes 'a pinch of salt'. If you want to standardize the recipe for a cookbook, you could determine that a 'pinch' is equivalent to, say, 1/8 teaspoon. By replacing 'a pinch of salt' with a specific measurement, you are effectively turning a vague idea into a precise instruction, much like how we eliminate arbitrary constants to form a clear PDE.

Given: z=ax+by+c

Step 1: Differentiate partially with respect to x and y: ∂z/∂x = a, ∂z/∂y = b

Let: ∂z/∂x = p, ∂z/∂y = q

So: p=a, q=b Substitute into original equation: z = px + qy + c ⇒ c = z − px − qy Now eliminate c: No constants left to eliminate further. Hence, the PDE is: ∂²z/∂x∂y = 0

In this example, we start with a function that has constants a, b, and c. First, we differentiate the function z = ax + by + c with respect to x and y. This gives us expressions for p and q, which are equal to the constants. Next, we substitute these expressions back into the original function to isolate c. Finally, since there are no constants left to eliminate, we can express the partial differential equation as ∂²z/∂x∂y = 0, which signifies that there is a relationship between the rates of change of z with respect to both variables.

Think of this process like simplifying an equation in a math class. If you're trying to solve for x in an equation like 'x + 5 = 10', you first isolate x by subtracting 5 from both sides. Here, we're doing a similar task—isolating constants to help reveal the underlying relationships in the PDE.

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

When we deal with functions that contain arbitrary functions instead of constants, the process is a bit different. We first differentiate the function with respect to the independent variables, which allows us to express those arbitrary functions in terms of their derivatives. After differentiating, we use the relationships obtained through differentiation to eliminate the arbitrary functions. This process helps in forming a new PDE that relates the variables and derivatives directly.

Consider trying to express an idea in writing without using some vague terms. For instance, instead of saying 'the fruit is good', you specify 'the banana is ripe'. By removing ambiguities, you make the message clearer. In a similar way, when we eliminate arbitrary functions through differentiation, we refine our equations to a clearer format that encapsulates the essential relationships within the PDE.

Given: z=f(x²+y²)

Let: u=x²+y², so z=f(u)

Differentiate: ∂z/∂x = f′(u)*2x ⇒ p = 2xf′(u) ∂z/∂y = f′(u)*2y ⇒ q = 2yf′(u)

Now, eliminate f′(u): Divide: p/q = f′(u) ⇒ (2x)/(2y) = (qx)/(py) This is the required PDE.

In this example, we started 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 u = x² + y², we differentiated z to express it in terms of p and q. After determining p and q, we utilized the relationships provided by these derivatives to eliminate f′(u). This leads us to the new partial differential equation that directly relates p and q without the arbitrary function present, simplifying our equation to its essentials.

Imagine trying to find the height of a tree based on its shadow. You could set up a relationship where you relate the length of the shadow to the height using geometry. Once you have a formula, you no longer need to refer to the actual setup (like the angle of the sun) directly because the formula encapsulates the necessary relationships. Similarly, we derive a PDE which encapsulates the relationship without reliance on the arbitrary function itself.

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