Discussion
For a homogeneous linear equation with constant coefficients, writing \( D = \partial/\partial x \) and \( D' = \partial/\partial y \), the auxiliary equation is formed by substituting
Answer: A
Putting \( D' = mD \) reduces the operator polynomial to one in \( m \), whose roots \( m_{1}, m_{2}, \dots \) give the solution as a sum of arbitrary functions:
\[ z \;=\; f_{1}(y + m_{1}x) + f_{2}(y + m_{2}x) + \cdots \]
Each root contributes one arbitrary function, in place of the arbitrary constant an ordinary equation would contribute
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