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How to Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$.

And then how would I show that the Wronskian of the two functions $f$ and $g$ is equal to zero, i.e. $W(f,g)=0$?

Also how would I establish that functions f and g are linearly independent on the interval $(-\infty, +\infty)$. Can a Wronksian be zero on all points and yet still be linearly independent?

Siminore
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Blue Pony Inc.
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2 Answers2

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Hints:

When you are doing the Wronskian with absolute values and trying to show linear independence, it is important to use the absolute value definitions. So, for the Wronskian, we would consider:

  • $t < 0 \rightarrow |t| = -t \rightarrow \text{Wronskian} = 0$ (show this)
  • $t = 0 \rightarrow |t| = 0 \rightarrow \text{Wronskian} = 0$ (show this)
  • $t > 0 \rightarrow |t| = t \rightarrow \text{Wronskian} = 0$ (show this)

Since the Wronskian is zero, no conclusion can be drawn about linear independence.

For linear independence, we want to go back to the basic definitions again. We have:

  • $|t| = t$ if $t \ge 0$ and $|t| = -t$ if $t \lt 0$. Thus, our equations become:

$$c_1 t^2 + c_2 t^2 = 0~~~~ \text{for}~ t \ge 0 \\ c_1t^2 - c_2 t^2 = 0~~~~\text{for}~ t \lt 0$$

What do you get for $c_1, c_2$ when you solve these simultaneous equations? What that does tell you about linear independence? Does a certain value of $t$ that makes these equations true matter?

Amzoti
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Recalling that we define the Wronskian $W(f,g)$ for two functions $f$ and $g$ on an interval $(a,b)$ as

$$ W(f,g) = \det\begin{bmatrix}f & g\\f' & g'\end{bmatrix} = fg' - gf'$$

we see that as on $\mathbb R$ we have that $f'(t)=2|t|$ and $g'(t) = 2t$ we see that $W(f,g)=0$ on $\mathbb R$.

However, while it is certainly true that if we have two linearly dependant equations, its Wronskian is $0$, the converse is certainly not true (at least without adding further conditions on $f$ and $g$), so from this we cannot conclude that the two functions are linearly dependant (and indeed they are not). How you could go about proving this, I'm slightly unsure.

Andrew D
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