Homogeneous linear differential equation of constant coefficients.

Question

Determine all real valued solutions of equation : $(D^3 -i D^2+D-i)y = 0 $ where $ D=d/dx $ and $i$ is iota.
The three roots are $i,±i$; and i don't how to proceed further. one of my doubt is that should I consider these roots as repeated roots or a conjugate pair of roots while solving.

Answer 1

You can go either way, because these operators all commute. Thus you can look at the operator as $(D+i)(D-i)^2$ or as $(D-i)(D^2+1)$.

That being said, they told you the solutions of interest are real-valued, and you can use that to avoid having to solve the entire equation at all, by noting that the imaginary part of the LHS must be zero, and realizing what that imaginary part looks like if $y$ is real-valued.