I need to know the general solutions ( and the assumptions ) to solve the following second order matrix differential equation.
$ AX'' + i BX' - C = 0$ where $X(t)$ is a vector in $\mathbb{R}^n$ for every $t \in [0, \infty ) $ and $A,B,C$ are $n\times n$ real matrices...and $X(0)= X_{o}$ is a given vector. I want to write down the solution $X(t)$ involving matrix exponentials. $A$ is invertible... I think it will also be necessary to assume that $A^{-1}C$ has a square-root..[ since if B is $0$, the solution is $X(t)= e^{t\sqrt{A^{-1}C}} ( X_{0}) $ ]...So we can assume $C$ is positive definite also... I know that the trick is to write it down as first order system and then solve and get back..but having a bit of trouble. Also, I need to know if symmetry of A, B and C is necessary or not..
I found a good reference for the question. M.Taylor's "Introduction to Differential Equation", chapter 3 (Linear systems of differential equations) , section 6 (Second order systems) discusses precisely these questions that I was looking for. Thanks