7.4 p. 395 #7
Consider the vectors $\mathbf{x}^{(1)}(t) = \begin{pmatrix}t^2\\2t\end{pmatrix}$ and $\mathbf{x}^{(2)}(t) = \begin{pmatrix}e^t\\e^t\end{pmatrix}$.
(a) Compute the Wronskian of $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$.
$$
W(\mathbf{x}^{(1)}(t),\mathbf{x}^{(2)}(t))=t^2 e^t - 2te^t = t(t - 2)e^t.
$$
(b) In what intervals are $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$ linearly independent?
When $t=0$ and $t=2$, $W(\mathbf{x}^{(1)}(t),\mathbf{x}^{(2)}(t))$ = 0 and $\mathbf{x}^{(1)}(t)$ and $\mathbf{x}^{(2)}(t)$ are linearly dependent.
so $\mathbf{x}^{(1)}(t)$ and $\mathbf{x}^{(2)}(t)$ and $\mathbf{x}^{(2)}(t)$ and $\mathbf{x}^{(2)}(t)$ are linearly independent at each point except when $t = 0$ and $t = 2$.
(c) What conclusion can be drawn about the coefficients in the system of homogeneous differential equations satisfied by $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$?
If $\mathbf{x}$ satisfies this system $\mathbf{x}'+A\mathbf{x}=0$ then $A$ must be singular at $t=0$ and $t=2$.
