How to reduce a system of second order ODEs to four first order equations?

[Bruno Tolentino]

Someone can explain me how to get the general solution for this system of ODE of second order with constant coeficients:$$\begin{bmatrix}<br /> a_{11} & a_{12}\\ <br /> a_{21} & a_{22}\\<br /> \end{bmatrix}<br /> <br /> \begin{bmatrix}<br /> \frac{d^2x}{dt^2}\\ <br /> \frac{d^2y}{dt^2}\\<br /> \end{bmatrix}<br /> <br /> +<br /> <br /> \begin{bmatrix}<br /> b_{11} & b_{12}\\ <br /> b_{21} & b_{22}\\<br /> \end{bmatrix}<br /> <br /> \begin{bmatrix}<br /> \frac{dx}{dt}\\ <br /> \frac{dy}{dt}\\<br /> \end{bmatrix}<br /> <br /> +<br /> <br /> \begin{bmatrix}<br /> c_{11} & c_{12}\\ <br /> c_{21} & c_{22}\\<br /> \end{bmatrix}<br /> <br /> \begin{bmatrix}<br /> x\\ <br /> y\\<br /> \end{bmatrix}<br /> <br /> =<br /> <br /> \begin{bmatrix}<br /> 0\\ <br /> 0\\<br /> \end{bmatrix}$$
OBS: source of the doubt: https://es.wikipedia.org/wiki/Movimiento_armónico_complejo

 

[deskswirl]

Reducing the order will give you 4 first order equations which is much easier to solve. In the the link you provided they are essentially making a similarity transformation (i.e. switching from the original coordinates to normal coordinates). You can find an extensive description of how and why this transformation is used in Boyca and DiPrima (Chapter 7?) or Coddington and Levinson (within the first 70 pages).