Question

Solve (2D^2D'-3DD'^2+D'^3)z=0

Answer 1

$$Solution$$

Given $(2D^2D'-3DD'^2+D'^3)z=0$

$D'(D^2-\frac32DD'+\frac12D'^2)z=0\\ D'(D^2-\frac12DD'-DD'+\frac12D'^2)z=0 \implies D'(D(D-\frac12D')-D'(D-\frac12D))\\ \therefore (D'(D-D')(D-\frac12D'))z=0$

Then

$D'\implies\phi_1(y)\\ (D-D')\implies\phi_2(y+x)\\ (D-\frac12D')\implies\phi_3(y+\frac12x)\\ z= \phi_1(y)+\phi_2(y+x)+\phi_3(y+\frac12x)\\$

Answer

$$z= \phi_1(y)+\phi_2(y+x)+\phi_3(y+\frac12x)\\$$