Question

(D3 − 3DD′2 + 2D′3)z = 0

Answer 1

 The auxillary equation is

$$m^3-3m+2=0$$

$$m^3-m-2m+2=0$$

$$m(m^2-1)-2(m-1)=0$$

$$(m-1)(m^2+m-2)=0$$

$$(m-1)^2(m+2)=0$$

The roots of the auxiliary equation are $1,1,-2.$

The solution is given by

$$z=f(y+x)+xg(y+x)+h(y-2x)$$