The equation in the image appears to be a system of differential equations that relates three variables $x$, $y$, and $z$ through differentials. It is written as:

$\frac{dx}{x^2 + y^2} = \frac{dy}{2xy} = \frac{dz}{(x + y)^3 z}$

To solve such a system of differential equations, one approach is to break it down into two separate equations and solve them step by step. Here’s how we can approach it:

Step 1: Compare the first two fractions

$\frac{dx}{x^2 + y^2} = \frac{dy}{2xy}$

This gives us a relationship between $x$ and $y$. Cross-multiply to eliminate the fractions:

$(x^2 + y^2) dy = 2xy \, dx$

This is a first-order differential equation in terms of $x$ and $y$. We can try various methods such as separation of variables or substitution depending on the simplification that arises.

Step 2: Compare the second and third fractions

$\frac{dy}{2xy} = \frac{dz}{(x + y)^3 z}$

This provides another equation that relates $y$ and $z$. Again, we can cross-multiply:

$2xy \, dz = (x + y)^3 z \, dy$

This gives us a separable differential equation in terms of $y$ and $z$, which can be solved similarly.

Next Steps

1. Solve the equation relating $x$ and $y$.
2. Substitute the result into the equation relating $y$ and $z$ to find a solution for $z$.
3. Apply initial conditions if provided to find particular solutions.

Would you like me to proceed with solving these equations step by step, or would you prefer additional explanations on specific parts of this system?

Further Exploration Questions:

1. How would the system change if we replaced the function $(x + y)^3$ with another power of $x + y$?
2. What methods can be used to solve nonlinear differential equations like the one for $x$ and $y$?
3. Can this system be expressed in terms of parametric equations for easier solving?
4. How would initial conditions influence the form of the solution for $z$?
5. Could these equations represent a physical system, and if so, which one?

Tip: When solving systems of differential equations, finding symmetries or using substitution can often simplify the process.