- #1

- 43

- 0

## Homework Statement

Given the two functions:

[itex]f(t) = t[/itex]

[itex]g(t) = |t|[/itex]

Use the Wronskian to determine if the two functions are dependent or independent.

**2. The attempt at a solution**

I have already found the correct answer to this, which is that it is independent but I have some questions as to how this is. When I first tried to solve this I found that it was dependent, based on the following reasoning:

[itex]W[f(t),g(t)] = \left| {\begin{array}{cc}

t & |t| \\

1 & \pm 1 \\

\end{array} } \right|

[/itex]

Calculating the determinant:

[itex] W[f(t),g(t)] = (t)(\pm 1) - (|t|)(1)[/itex]

[itex] W[f(t),g(t)] = (t)(\pm 1) - |t|[/itex]

Given that [itex]t<0[/itex] and [itex]0<t[/itex] will determine the sign of [itex]g'(t) = \frac{d}{dt} |t|[/itex], then utilizing the following conditions:

If [itex]t = +1[/itex], then [itex]g'(t) = +1[/itex].

If [itex]g'(t) = +1[/itex], then:

[itex] W[f(t),g(t)] = (t)(\pm 1) - |t|[/itex]

[itex] W[f(1),g(1)] = (1)(+1) - |1| = 0[/itex]

Likewise, if [itex]t = -1[/itex], then [itex]g'(t) = -1[/itex].

If [itex]g'(t) = -1[/itex], then:

[itex] W[f(t),g(t)] = (t)(\pm 1) - |t|[/itex]

[itex] W[f(-1),g(-1)] = (-1)(-1) - |-1| = 1 - 1 = 0[/itex]

Thus, based on the above reasoning, I thought the answer was dependent, but in fact, it is independent; can someone point out my mistake?