# Converting second order system into first order system (ODE)

The second order equation

$\frac{d^2\vec{x}}{dt^2} = A\vec{x}\ + \vec{g}(t)$

models an earthquake's effect on a 7-story building. Let $x_j(t)$ be the displacement of the $j$th floor with respect to its equilibrium position. The ground moves with displacement $g(t)$.

Here

$\vec{x} = \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_7 \end{pmatrix}$

$\vec{g}(t) = \begin{pmatrix} g(t)\\ 0\\ \vdots\\ 0 \end{pmatrix}$ .

A second order $7\times7$ system in $x_j(t)$ is given by

• $x_1'' = 10(x_2- x_1- 1)$
• $x_2'' = 10(x_3- 2x_2+ x_1)$
• $x_3'' = 10(x_4- 2x_3+ x_2)$
• $x_4'' = 10(x_5- 2x_4+ x_3)$
• $x_5'' = 10(x_6- 2x_5+ x_4)$
• $x_6'' = 10(x_7- 2x_6+ x_5)$
• $x_7'' = 10(x_6- x_7)$.

Write the above second order system as a first order $14\times14$ system using the additional equations $v_j = x'_j$.