1.2.2. Example 4

Given: z=f(x+y)+g(x−y) Let: u=x+y, v=x−y⇒z=f(u)+g(v)

This chunk introduces a mathematical function z that depends on two new variables, u and v. Here, u is defined as (x+y) and v as (x−y). The function z is represented as a sum of two other functions, f and g, which depend on these variables u and v respectively.

Think of z as a recipe that combines two ingredients (u and v). Each ingredient (the functions f and g) contributes uniquely to the overall flavor (the function z). Just like different combinations of ingredients can create new dishes, different combinations of u and v will change the outcome of z.

Differentiate: ∂z ∂z =f′(u)+g′(v), =f′(u)−g′(v) ∂x ∂y

In this step, we take the partial derivatives of z with respect to x and y. The partial derivative with respect to x leads to f'(u) + g'(v), while the derivative with respect to y gives us f'(u) - g'(v). This mathematical manipulation helps to find how z changes as we slightly change x or y, which is crucial for forming a PDE.

Imagine monitoring the temperature of a mixture as you add different ingredients. The way the temperature changes when you add each ingredient is like taking a derivative. It tells you how sensitive the overall result (temperature) is to changes in each of the ingredients (x and 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

This chunk elaborates on the manipulation of the derivatives we calculated earlier. By adding and subtracting these equations, we can isolate the derivatives of f and g. The results show that f' and g' can be defined in simpler forms, expressed in terms of p and q.

Consider a situation where you’re measuring the height of a plant from two different perspectives. If you add the heights from both perspectives, you get a combined view. By doing so, you can also find out the individual heights (f’ and g’) of the different parts of the plant just like we’re finding those derivatives here.

Now differentiate again and eliminate f and g to form a second-order PDE.

Finally, we take one more derivative step, which allows us to eliminate the arbitrary functions f and g using the relationships we've established. This leads us toward a second-order Partial Differential Equation (PDE) that describes the changes in z based on the variables x and y without dependence on arbitrary functions.

Imagine you’ve gathered enough data on a plant’s growth (the function z) that you no longer need to refer to detailed observations (the functions f and g). You’ve formed a general rule (the PDE) that predicts future growth based on the current conditions, simplifying your analysis.

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