# (Real) Solutions for Second Order Differential Equations

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?

According to Undetermined Coefficients— Annihilator Approach; for the second one assume $$y_p(t)=A\sin(2t)+B\cos(2t)$$ for some proper unknown constants $A$ and $B$, then substitute it in the ODE and find the constants. Your attempt to find them was correct, $A=0,B=-1/3$
The solution of the second equation is correct. For the first-one there is no real solution other than the trivial solution ($y=0$) since the coefficient involves a complex number(The solution will depend on the coefficients).