Chapter 6.1, Problem 14P

$(a)$ Using matrix notation, show that the system of second order equations $(18)$ for the displacement $y={(y_1,y_2)}^T$ of the masses can be written in the form

$y^{"}=Ky+f(t)$.

$(b)$ Under what conditions on $k_1$, $k_2$, $k_3$, $m_1$, and $m_2$ is $K$ a symmetric matrix?

$m_1\frac{d^2{y}_1}{d{t}^2}=k_2(y_2-y_1)-k_1{y}_1+F_1(t)$

$=-\left(k_1+k_2\right)y_1+k_2{y}_2+F_1\left(t\right)$ $(18)$

$m_2\frac{d^2{y}_2}{d{t}^2}=-k_3{y}_2-k_2(y_2-y_1)+F_2(t)$

$=k_2{y}_1-(k_1+k_3)y_2+F_2(t)$.