(Spotted by Thomas Hsu)
A numerical error has crept in Equation (4)
on page 377—the eigenvectors are incorrect
as shown. The corrected version of the equation is:
x_h(t) =
\begin{bmatrix}
x_1(t)\\x_2(t)
\end{bmatrix}
=
c_1 e^{-0.025 t}
\begin{bmatrix}
1 \\ 1.732
\end{bmatrix}
+
c_2 e^{-0.095 t}
\begin{bmatrix}
1 \\ -1.732
\end{bmatrix}.
Equation (5) on page 378 needs to be corrected accordingly.
Upon applying the initial conditions we find that
c_1 = -26.93 and c_2 = 1.93. Finally, we redo the graphs in
Figure 6.13 on page 378 and we get:
Aside: If instead of decimal points we use true fractions in
the computations, we see that the eigenvalues and eigenvectors, and
the coefficients c_1 and c_2 are:
\lambda = \frac{1}{50} \big[ -3 \pm \sqrt{3} \big],
\quad
\mathbf{v}
=
\begin{bmatrix}
1 \\ \pm \sqrt{3}
\end{bmatrix},
\quad
c_1 = -25\Big(\frac{1}{2} + \frac{1}{3} \sqrt{3}\Big),
\quad
c_2 = -25\Big(\frac{1}{2} - \frac{1}{3} \sqrt{3}\Big).