Using Lagrange’s method, solve the differential equation:

To solve the given differential equation using Lagrange's method, we assume that the solution can be expressed in the form:

\[p(x,y,z) = \frac{\partial S}{\partial x}, \quad q(x,y,z) = \frac{\partial S}{\partial y}\]

where \(S(x,y,z)\) is a certain function to be determined.

Then, we substitute these expressions into the given equation:

\[ (x^2 - y^2 - z^2) \frac{\partial S}{\partial x} + 2x \frac{\partial S}{\partial y} = _________ ___ ________ _______ _____ ____ ________ _______.
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