To form a partial differential equation (PDE) by eliminating the arbitrary functions $f$ and $g$ from the given expression $z = x f(x+y) + g(x+y)$, follow these steps:

Step 1: Differentiate with respect to $x$ and $y$

1. Differentiate with respect to $x$: $\frac{\partial z}{\partial x} = f(x+y) + x f'(x+y) \cdot \frac{\partial (x+y)}{\partial x} + g'(x+y) \cdot \frac{\partial (x+y)}{\partial x}$ Simplifying: $\frac{\partial z}{\partial x} = f(x+y) + x f'(x+y) + g'(x+y)$

2. Differentiate with respect to $y$: $\frac{\partial z}{\partial y} = x f'(x+y) \cdot \frac{\partial (x+y)}{\partial y} + g'(x+y) \cdot \frac{\partial (x+y)}{\partial y}$ Simplifying: $\frac{\partial z}{\partial y} = x f'(x+y) + g'(x+y)$

Step 2: Form a system of equations

Now, we have two expressions:

1. $\frac{\partial z}{\partial x} = f(x+y) + x f'(x+y) + g'(x+y)$
2. $\frac{\partial z}{\partial y} = x f'(x+y) + g'(x+y)$

Step 3: Eliminate the arbitrary functions

Subtract the second equation from the first: $\frac{\partial z}{\partial x} - \frac{\partial z}{\partial y} = f(x+y)$

Now, differentiate the resulting equation with respect to $y$ to eliminate $f(x+y)$: $\frac{\partial^2 z}{\partial y \partial x} - \frac{\partial^2 z}{\partial y^2} = f'(x+y)$

Using the second original equation to replace $x f'(x+y) + g'(x+y)$ back, we can eliminate $f'(x+y)$.

Finally, combine and differentiate appropriately to eliminate $g(x+y)$ and form the desired PDE.

The resulting PDE will involve $z$, $\frac{\partial z}{\partial x}$, $\frac{\partial z}{\partial y}$, and their higher-order derivatives.

Would you like to go into more details on the final steps of forming this PDE?

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Relative questions:

1. How do you interpret the roles of arbitrary functions in the formation of PDEs?
2. What other techniques are there for eliminating arbitrary functions in PDEs?
3. Can you explore examples of similar PDE formations with different initial functions?
4. How does the presence of mixed partial derivatives affect the structure of the PDE?
5. What are the potential solutions or general forms for such PDEs?

Tip: Always verify if the resulting PDE is consistent by checking if it satisfies the original function when substituting back.