Consider vectors
$$
x^{(1)}(t) = \left(
\begin{array}{cccc}
t \\
1 \\
\end{array}
\right) \
$$
$$
x^{(2)}(t) = \left(
\begin{array}{cccc}
e^t \\
e^t \\
\end{array}
\right) \
$$
(a) Compute the Wronskian of $x^{(1)},x^{(2)}$
(b) In what intervals are $x^{(1)},x^{(2)}$ linearly independent?
(c) Can be solutions $\left( x^{(1)},x^{(2)} \right)$ for some homogeneous linear differential equation:
$x' = A(t).x$ where $A(t)$ is continuous on $R$ ? Explain your answer.
For part (a) Wronskian is $e^t (t - 1)$
For (b) the Wronskian is 0 when $t=1$ so the interval is $(-∞,1)(1,∞)$
I am confused for (c) since $\left( x^{(1)},x^{(2)} \right)$ is not linearly independent at $t=1$, so can we say they are solutions only when t is not equal to 1 ?
It depends on your initial conditions. So the Wronskian is zero if your initial condition is for example $x^{1}(1)=x_{1}$ and $x^{2}(1)=x_{2}$ (see Abel's formula). In fact the $W(x^{1},x^{2})(t)=0$ for all time or $W(x^{1},x^{2})(t)\neq 0$ for all time.