Find the fundamental solutions of the following differential operators. Check that they satisfy (outside the singularities) the homogeneous equation in principal variables and the conjugate one in dual variables $$\frac{\partial ^2}{\partial t^2}-\frac{\partial ^2}{\partial x^2}+2\frac{\partial ^2}{\partial y \partial t}+2\frac{\partial ^2}{\partial z\partial t}-2\frac{\partial ^2}{\partial y\partial z}$$
My attempt:
I thought that in order to find the fundamental solutions, you have to solve the equation $$\frac{\partial ^2u}{\partial t^2}-\frac{\partial ^2u}{\partial x^2}+2\frac{\partial ^2u}{\partial y \partial t}+2\frac{\partial ^2u}{\partial z\partial t}-2\frac{\partial ^2u}{\partial y\partial z}=0$$
It can be seen that the solution will be in the form of separable functions, i.e. $u(x, y, z, t) = X(x)Y(y)Z(z)T(t)$, then
$$\frac{X''}{X}=\frac{Y''}{Y}+2\frac{T'}{T}\frac{Y'}{Y}+2\frac{T'}{T}\frac{Z'}{Z}-2\frac{Y'}{Y}\frac{Z'}{Z}+\frac{T''}{T}$$
I was able to get to this point, but then I don't know how to solve it. Am I using the right method, is there no simpler method of solving it?