Coupled system of linear second order differential equations I desperately need to solve a coupled system of linear second order differential equations of the form:
$$x''+ax'+bx-cy'-dy=0$$
$$y''+ay'+by+cx'+dx=0$$
where both "x" and "y" are functions of time and {a,b,c,d} are constants, I need a method other than that of Laplace transform, any suggestions?
Edit: Is it possible to solve this system by means of "matrix exponential"?
Is there any textbook on differential equations that covers solving this sort of systems?
 A: The most general approach to these problems is to write your system as a four dimensional first order ODE system:
\begin{align}
 x' &= \xi\\
 \xi' &= - b x - a \xi +d y + c \eta\\
 y ' &= \eta\\
 \eta' &= -d x -c \xi -b y -a \eta
\end{align}
which can be written in matrix form
\begin{equation}
\mathbf{x}' = A\, \mathbf{x}
\end{equation}
with
\begin{equation}
 \mathbf{x} = \begin{pmatrix} x \\ \xi \\ y \\ \eta \end{pmatrix}\qquad \text{and} \qquad A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ -b & -a & d & c \\ 0 & 0 & 0 & 1 \\ -d & -c & -b & -a \end{pmatrix}.
\end{equation}
Surprisingly, the eigenvalues of this matrix are not particularly hard to find, as are the eigenvectors. The general solution to the above system is then of the form
\begin{equation}
 \mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{x}_1 + c_2 e^{\lambda_2 t}\mathbf{x}_2 + c_3 e^{\lambda_3 t}\mathbf{x}_3 + c_4 e^{\lambda_4 t}\mathbf{x}_4,
\end{equation}
where $\lambda_{1,2,3,4}$ are the eigenvalues of the matrix $A$ and $\mathbf{x}_{1,2,3,4}$ are the associated eigenvectors. The constants $c_{1,2,3,4}$ are determined by the initial conditions.
As for your question for a literature reference: I can't imagine a textbook on differential equations not treating the above approach, so pick your favourite.
