I need help with the first part mostly!

The Question:

for the associated homogeneous

D = d/dx

roots a = b = i

Trial solution:

when you sub it into the homogeneous, the real parts cancel but the imaginary doesn't!

I was wondering why this is?

Am i right in thinking the trial solution is wrong?

For the particular solutions of the other parts I got

for

I got

for

I got

please verify these!

thanks