To solve question 5 (i), we need to derive the partial differential equation arising from the given surface:

Given:

$z = xy + f(x^2 + y^2)$

Here $f$ is a function of $x^2 + y^2$.

Solution:

1. Step 1: Differentiate partially with respect to $x$: $\frac{\partial z}{\partial x} = y + f'(x^2 + y^2) \cdot \frac{\partial (x^2 + y^2)}{\partial x}$ Since $x^2 + y^2$ is a function of $x$, $\frac{\partial (x^2 + y^2)}{\partial x} = 2x$ So, $\frac{\partial z}{\partial x} = y + 2x f'(x^2 + y^2)$

2. Step 2: Differentiate partially with respect to $y$: $\frac{\partial z}{\partial y} = x + f'(x^2 + y^2) \cdot \frac{\partial (x^2 + y^2)}{\partial y}$ Since $x^2 + y^2$ is a function of $y$, $\frac{\partial (x^2 + y^2)}{\partial y} = 2y$ So, $\frac{\partial z}{\partial y} = x + 2y f'(x^2 + y^2)$

3. Step 3: Eliminate $f'$: From the expressions for $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$: $\frac{\partial z}{\partial x} - y = 2x f'(x^2 + y^2)$ $\frac{\partial z}{\partial y} - x = 2y f'(x^2 + y^2)$

   Dividing these equations: $\frac{\frac{\partial z}{\partial x} - y}{\frac{\partial z}{\partial y} - x} = \frac{2x}{2y}$ Simplify: $\frac{\frac{\partial z}{\partial x} - y}{\frac{\partial z}{\partial y} - x} = \frac{x}{y}$

4. Step 4: Rearrange into the required partial differential equation: Cross-multiply to get: $y \left(\frac{\partial z}{\partial x} - y\right) = x \left(\frac{\partial z}{\partial y} - x\right)$ Expand: $y \frac{\partial z}{\partial x} - y^2 = x \frac{\partial z}{\partial y} - x^2$ Rearrange: $y \frac{\partial z}{\partial x} - x \frac{\partial z}{\partial y} = y^2 - x^2$

Final Answer:

The required partial differential equation is: $y \frac{\partial z}{\partial x} - x \frac{\partial z}{\partial y} = y^2 - x^2$

Let me know if you would like detailed clarifications or have further questions!

Related Questions:

1. How do you derive higher-order partial differential equations for similar surfaces?
2. What happens if $f(x^2 + y^2)$ is specified as a concrete function?
3. How do you solve the resulting PDE if boundary conditions are given?
4. Can you generalize the process for other surfaces like $z = g(x^2 + y^2)$?
5. What are the physical interpretations of such PDEs in applied mathematics?

Tip: Always ensure all derivatives are expressed in terms of given variables to avoid errors in elimination processes.