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To solve the given first-order partial differential equation (PDE) using Lagrange's method, let's first rewrite the equation: \[ (x^2 - y^2 - z^2)p + 2xq = 2xz \] Where: - \( p = \frac{\partial z}{\partial x} \), - \( q = \frac{\partial z}{\partial y} \). ### Step 1: Identify the form of the PDE The PDE is of the form: \[ A(x, y, z) \frac{\partial z}{\partial x} + B(x, y, z) \frac{\partial z}{\partial y} = C(x, y, __________ _________ ___ ______ __________ ______ ________ _______ _______ __________ _______.
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