# System of ODE of second order

1. Jul 29, 2015

### Bruno Tolentino

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

2. Jul 29, 2015

### 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).

Last edited: Jul 29, 2015