Word Count : 512

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The given family of surfaces is represented as: \[ \phi(x + y + z, x^2 + y^2 - z^2) = 0 \] This equation involves two variables, \( u_1 = x + y + z \) and \( u_2 = x^2 + y^2 - z^2 \), where the function \(\phi(u_1, u_2) = 0\) defines the family of surfaces. To find the differential equation of this family of surfaces, we need to differentiate the equation with respect to the independent variables (here, \(x\), \(y\), and \(z\)) and eliminate the function \(\phi\). ### Step 1: Implicit differentiation We will differentiate \(\phi(x + y + z, x^2 + y^2 - z^2) = 0\) implicitly with respect to the independent variables \(x\), \(y\), and \(z\). Let: \[ u_1 = x + y ______ _____ _______ ____ ___ __________ __________ ______ ___.
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