I want to convert the coupled differential equation to a system of first-order differential equations, Is it possible. Can anybody help me?
$$\begin{bmatrix} A & B\\ B & E \end{bmatrix} \begin{bmatrix}\ddot{y}\\ \ddot{x} \end{bmatrix}+\begin{bmatrix}C & 0\\0&F \end{bmatrix}\begin{bmatrix}{y}\\ {x} \end{bmatrix}=\begin{bmatrix}D\\ G \end{bmatrix}$$ where $y$ and $x$ are the dependent variables.
Hint: Introduce the variables $x = x_1$, $\dot{x}=\dot{x}_1=x_2$ and $y = x_3$, $\dot{y}=\dot{x}_3=x_4$.
Your equations are
$$\dot{x}_1=x_2$$ $$\ddot{x}=\dot{x}_2=?$$ $$\dot{x}_3=x_4$$ $$\ddot{y}=\dot{x}_4=?$$
can you fill in the remaining equations?
As you system looks like a mechanical mass-spring system
$$\boldsymbol{M\ddot{q}}(t)+\boldsymbol{Kq}(t)=\boldsymbol{F}(t)$$
you can also use modal decomposition to turn the coupled second order system to an uncoupled second order system.
You will have to determine the eigenfrequencies $\omega_i$ by solving
$$\det\left[-\omega^2\boldsymbol{M}+\boldsymbol{K} \right]=0$$
and the eigenvectors $\hat{\boldsymbol{q}}$ by solving
$$\left[-\omega^2\boldsymbol{M}+\boldsymbol{K} \right]\hat{\boldsymbol{q}}=\boldsymbol{0}$$
In order to determine the uncoupled system, you can use the procedure from this question.