Solve two coupled second order differential equations using ode45 in MatLab?

Question

How to solve two coupled second order differential equations using ode45 in MatLab?

Equations are:

$b_1\cdot\ddot{X}+b_2\cdot\ddot{Y}+b_3\cdot X+b_4\cdot Y+b_5\cos{2t}\cdot X=0$

$a_1\cdot\ddot{X}+a_2\cdot\ddot{Y}+b_4\cdot X+a_3\cdot Y=0$

where $t$ is time variable, overdots are time derivatives and $a$ and $b$ are constants. Notice the two coupled second order derivatives in both equations.

Answer 1

Let $A=\begin{bmatrix} b_1 & b_2 \\ a_1 & a_2 \end{bmatrix}$. Then $\begin{bmatrix} \ddot{X} \\ \ddot{Y} \end{bmatrix} = -A^{-1} \begin{bmatrix} b_3 X + b_4 Y + b_5 \cos(2t) X \\ b_4 X + a_3 Y \end{bmatrix}$. You can combine this with new variables $B=\dot{X},C=\dot{Y}$ to get a system of four scalar first order differential equations.

This assumes $A$ is invertible in the first place; the system is ill-defined otherwise.