Chapter 4, Problem 1RE

To determine

Whether the function $S=\left\{e^{5t},1\right\}$ is linearly independent or linearly dependent.

Answer to Problem 1RE

The function $S=\left\{e^{5t},1\right\}$ is linearly independent.

Explanation of Solution

Procedure used:

1. Calculate the wronskian as,

$W\left(s\right)=\left|\begin{array}{cc}{y}_1\left(t\right)& y_2\left(t\right)\\ \frac{d}{dt}\left(y_1\left(t\right)\right)& \frac{d}{dt}\left(y_2\left(t\right)\right)\end{array}\right|$

2. If the wronskian is zero, then the function is linearly dependent.

3. If the wronskian is not zero, then the function is linearly independent.

Calculation:

Consider the function.

$S=\left\{e^{5t},1\right\}$

Here, $y_1=e^{5t}$ and $y_2=1$

The wronskian is,

$\begin{array}{c}W\left(s\right)=\left|\begin{array}{cc}{e}^{5t}& 1\\ \frac{d}{dt}\left(e^{5t}\right)& \frac{d}{dt}\left(\left(1\right)\right)\end{array}\right|\\ =\left|\begin{array}{cc}{e}^{5t}& 1\\ 5e^{5t}& 0\end{array}\right|\\ =e^{5t}\left(0\right)-5e^{5t}\left(1\right)\\ =-5e^{5t}\end{array}$

The wronskian is not zero.

Therefore, the function $S=\left\{e^{5t},1\right\}$ is linearly independent.