Chapter 9.11, Problem 33AP

Consider the differential equation

$\frac{d^{3}y}{d{t}^3}+a\frac{d^{2}y}{d{t}^2}+b\frac{dy}{dt}+cy=0$

where $a$, $b$ and $c$ are arbitrary functions of $t$.

(a) Replace Equation $\left(\mathrm{9.11.8}\right)$ by an equivalent linear system $x^{\prime }=Ax$ for an appropriate $3\times 3$matrix $A$.

(b) If $y_1=f_1\left(t\right)$, $y_2=f_2\left(t\right)$, and $y_3=f_3\left(t\right)$ are solutions to the system you derived in part (a) are $x_1\left(t\right)=\left[\begin{array}{c}{f}_1\left(t\right)\\ {f^{\prime }}_1\left(t\right)\\ {f^{″}}_1\left(t\right)\end{array}\right]$, $x_2\left(t\right)=\left[\begin{array}{c}{f}_2\left(t\right)\\ {f^{\prime }}_2\left(t\right)\\ {f^{″}}_2\left(t\right)\end{array}\right]$, $x_3\left(t\right)=\left[\begin{array}{c}{f}_3\left(t\right)\\ {f^{\prime }}_3\left(t\right)\\ {f^{″}}_3\left(t\right)\end{array}\right]$.

(c) Show that $W\left[x_1,x_2,x_3\right]\left(t\right)=W\left[y_1,y_2,y_3\right]\left(t\right)$