The given equation in the image is:

$\left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2 = 2$

This is a partial differential equation, where $z = z(x, y)$. The goal is to solve for $z$, given this equation. Let's analyze it:

1. We are given the sum of squares of partial derivatives of $z$ with respect to $x$ and $y$, equal to 2.

2. This resembles the equation of a circle in two dimensions, but in the context of partial derivatives, it describes level curves or surfaces in a certain way.

To solve this equation, you can attempt to separate the variables or apply methods such as characteristics, or trial solutions based on symmetry, like assuming a specific form for $z(x, y)$.

Would you like me to proceed with a particular solution approach (e.g., assuming $z(x, y)$ is a simple function like linear or quadratic), or explore further details of how to approach this problem?

Here are five follow-up questions:

1. Are there boundary conditions or initial values provided for $z(x, y)$?
2. Should we explore trial solutions for $z(x, y)$ (e.g., functions like polynomials)?
3. Would you like a step-by-step method of solving this partial differential equation (PDE)?
4. Is this part of a larger problem involving geometry or physics (e.g., wave or heat equations)?
5. Do you want to explore numerical methods for solving PDEs?

Tip: In many PDE problems, trying an assumption or symmetry in the solution (e.g., assuming $z(x, y)$ is a product of single-variable functions) can simplify the process significantly.