We are given the equation $z = f(x) + g(y)$, where $f(x)$ and $g(y)$ are arbitrary functions of $x$ and $y$, respectively. We need to find the partial differential equation by eliminating the arbitrary functions.

Step 1: Compute the first partial derivatives

Let's calculate the partial derivatives of $z$ with respect to $x$ and $y$.

1. Partial derivative with respect to $x$: $\frac{\partial z}{\partial x} = f'(x)$ where $f'(x)$ denotes the derivative of $f$ with respect to $x$.

2. Partial derivative with respect to $y$: $\frac{\partial z}{\partial y} = g'(y)$ where $g'(y)$ denotes the derivative of $g$ with respect to $y$.

Step 2: Compute the second mixed partial derivative

Now, let's compute the second mixed partial derivative of $z$.

1. Second mixed partial derivative: $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right) = \frac{\partial}{\partial y} \left( f'(x) \right) = 0$ Since $f'(x)$ depends only on $x$, its derivative with respect to $y$ is zero.

   Similarly, we can show that: $\frac{\partial^2 z}{\partial y \partial x} = 0$

Step 3: Form the partial differential equation

The second mixed partial derivative of $z$ is zero, leading to the following partial differential equation: $\frac{\partial^2 z}{\partial x \partial y} = 0$

This is the required partial differential equation.

Would you like further details or have any questions?

5 Related Questions:

1. What are the general solutions of this partial differential equation?
2. How would the PDE change if $z = f(x)g(y)$?
3. Can arbitrary constants be eliminated similarly to arbitrary functions in PDEs?
4. What happens if $f(x)$ and $g(y)$ are specific known functions instead of arbitrary?
5. How do boundary conditions affect the solution of this PDE?

Tip:

When dealing with arbitrary functions in PDEs, always compute the mixed partial derivatives to check if they can be eliminated. This often simplifies the process.