1. We are asked to solve the partial differential equation (PDE): $$(z^2 - 2yz - y^2)p + (xy + zx)q = xy - zx.$$ 2. Here, $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$. 3. The PDE can be written as: $$ (z^2 - 2yz - y^2)\frac{\partial z}{\partial x} + (xy + zx)\frac{\partial z}{\partial y} = xy - zx. $$ 4. To solve, we use the method of characteristics. The characteristic equations are: $$ \frac{dx}{z^2 - 2yz - y^2} = \frac{dy}{xy + zx} = \frac{dz}{xy - zx}. $$ 5. We now analyze the system: - From the first two fractions: $$ \frac{dx}{z^2 - 2yz - y^2} = \frac{dy}{xy + zx}. $$ 6. To simplify, factor and write the terms clearly, but note that these expressions are nonlinear and coupled. 7. Similarly, from the last two fractions: $$ \frac{dy}{xy + zx} = \frac{dz}{xy - zx}. $$ 8. We must integrate these ODEs along characteristic curves. --- 9. For the second PDE: $$ (3x + y - z)p + (x + y - z)q = 2(z - y). $$ 10. Rewriting: $$ (3x + y - z)\frac{\partial z}{\partial x} + (x + y - z)\frac{\partial z}{\partial y} = 2(z - y). $$ 11. The characteristic equations are: $$ \frac{dx}{3x + y - z} = \frac{dy}{x + y - z} = \frac{dz}{2(z - y)}. $$ 12. To solve, we consider the system: - From the first two ratios: $$ \frac{dx}{3x + y - z} = \frac{dy}{x + y - z}. $$ 13. From the second two ratios: $$ \frac{dy}{x + y - z} = \frac{dz}{2(z - y)}. $$ 14. Solve these ODEs along the characteristic curves to find implicit relations involving $x,y,z$. 15. The solution to each PDE is given implicitly by two independent constants obtained from integrating these characteristic systems.