The given partial differential equation is:

$(x^3 + 3xy^2)p + (y^3 + 3x^2y)q = 2(x^2 + y^2)z$

Here, $p = \frac{\partial z}{\partial x}$, $q = \frac{\partial z}{\partial y}$, and $z$ is the dependent variable, while $x$ and $y$ are the independent variables.

Approach:

This is a linear first-order partial differential equation. To solve it, let's follow these steps:

1. Check for exactness or separability: Rewrite the PDE to check if it can be solved using an exact method or separation of variables.
2. Find characteristic equations: We will use the method of characteristics to solve the PDE.
3. Solve along characteristic curves: Use the relationships between $p, q, z, x, y$ to determine $z$.

Step 1: Rewrite and identify terms

The PDE can be rewritten in terms of its coefficients: $P(x, y, z)p + Q(x, y, z)q = R(x, y, z),$ where: $P = x^3 + 3xy^2, \quad Q = y^3 + 3x^2y, \quad R = 2(x^2 + y^2)z.$

Step 2: Characteristic equations

The characteristic equations are derived from: $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}.$ Substituting the expressions for $P$, $Q$, and $R$: $\frac{dx}{x^3 + 3xy^2} = \frac{dy}{y^3 + 3x^2y} = \frac{dz}{2(x^2 + y^2)z}.$

Step 3: Solve characteristic equations

1. First pair: $\frac{dx}{x^3 + 3xy^2} = \frac{dy}{y^3 + 3x^2y}$: Simplify the terms to find the relationship between $x$ and $y$. Observe that the symmetry in $P$ and $Q$ suggests a homogeneous function of degree 3.

   Dividing $P$ and $Q$ by $x^3$, we notice that: $\frac{dx}{x^3} = \frac{dy}{y^3}.$ Solving, we find: $\frac{x}{y} = k_1, \quad \text{(a constant)}.$

   Therefore, $x$ and $y$ are related as: $y = k_1 x.$

2. Second pair: $\frac{dx}{x^3 + 3xy^2} = \frac{dz}{2(x^2 + y^2)z}$: Using the substitution $y = k_1 x$, express $y$ in terms of $x$: $y = k_1 x, \quad y^2 = k_1^2 x^2.$ Substitute into the coefficients: $P = x^3 + 3x(k_1^2 x^2) = x^3(1 + 3k_1^2),$ $Q = y^3 + 3x^2y = k_1^3 x^3 + 3x^2(k_1 x) = k_1 x^3(1 + 3k_1).$ Substitute these into the characteristic equations to simplify and solve for $z$.

Step 4: General solution

The solution will take the form: $z = F\left(\frac{x}{y}\right) \cdot \exp\left(\int \frac{2(x^2 + y^2)}{x^3 + 3xy^2} dx\right),$ where $F$ is an arbitrary function determined from boundary conditions.

Would you like detailed integration or further clarification? Here are some related questions:

1. How can we simplify the integration for characteristic curves?
2. Would a numerical solution be helpful for arbitrary values of $k_1$?
3. Can we express $z$ explicitly in terms of $x$ and $y$ for specific boundary conditions?
4. Is there symmetry in the PDE that can simplify finding the solution?
5. Would an alternative method, like a power series expansion, be beneficial?

Tip: Always look for homogeneous functions or symmetry in the coefficients of a PDE, as they often suggest substitutions that simplify the solution process.