11.5.4.4 With constant coefficients

Problem: Find all the values of \( \beta \), for which there is a linear transformation of variables \( (x, y) \rightarrow(z, t) \), bringing the equation to the following canonical form: a) \( u_{t t}= \) \( =u_{z z} \), b) \( u_{t}=u_{z z} \). \[ u_{x x}+4 u_{x y}-\beta u_{y y}-\beta u_{x}+\beta^{2} u_{y}=0 . \]