I want to solve the two following differential equations:
(i) All real solutions for $ y''(t) - iy(t) = 0$
(ii) A particular solution for $y''(t)+y(t) = \cos(2t) $
My attempt for (i) is to use $y = e^{\lambda t} $, but then I arrive at $y(t) = \displaystyle{c_1e^{\frac{(1+i)t}{\sqrt(2)}} + c_2e^{\frac{(1-i)t}{\sqrt(2)}}}$ - but these are complex solutions - so how can I find the real solutions as asked for?
My attempt for (ii): I solve with y(t) = $e^{\lambda t}$, I (after a pretty long calculation) arrive at $y(t) = c_1\sin(t) + c_2\cos(t)-1/3\cos(2t)$. Is this correct?