Word Count : 751

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To find the integral surface of the given partial differential equation (PDE): \[ (x - y)p + (y - x - z)q = z \] where \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \), we can use the method of characteristics. Here's a step-by-step solution: ### Step 1: Rewrite the PDE in Symmetric Form The PDE can be represented as: \[ \frac{dx}{x - y} = \frac{dy}{y - x - z} = \frac{dz}{z} \] ### Step 2: Solve the First Pair of Equations Consider the first and third terms: \[ \frac{dx}{x - y} = \frac{dz}{z} \] This suggests a relationship between \( x \), \( y \), and \( z \). However, it's more straightforward to find two independent first integrals. ### Step 3: Find the First Integral Let’s solve the equation: \[ \frac{dx}{x - y} = \frac{dy}{y - x - z} = \frac{dz}{z} ___ ________ ____ _______ ____ _____ ______ ___.
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