Question

Solve the differential equation.
(D²+D+D')z=0

Answer 1

Let,$D=h,D'=k$ ,thus

$$(D²+D+D')z=0\cong h^2+h+k=0$$

Hence,

$$h_1=\frac{-1+\sqrt{1-4k}}{2}\\h_2=\frac{-1-\sqrt{1-4k}}{2}$$

Then

$$C.F=\sum C_1e^{h_1x+ky}+\sum C_2e^{h_2x+ky}$$