Linear Algebra – How to Determine Linear Dependence Using Wronskian

linear algebraordinary differential equations

Here is the theorem I use:

Two solutions $\phi_1$, $\phi_2$ of $L(y)=y''+a_1y'+a_2y=0$, where $a_1$ and $a_2$ are constants, are linearly independent on an interval $I$ if, and only if, the Wronskain $W(\phi_1,\phi_2)\ne0$ for all $x\in I.$

Then, consider the functions: $$\phi_1(x)=x,\phi_2(x)=|x|, x\in(-\infty,\infty). $$Are they linearly dependent ot independent?

My answer is that they are linearly dependent since when $x\ge0$, $\phi_1(x)=x=\phi_2(x)$ and I plug them and the corresponding derivatives into the Wronskian, $W=0$ for all $x\ge0$; and also check when $x\lt0$, the Wronskian also equals to 0, thus $\phi_1$ and $\phi_2$ here are linear dependent. But the answer in the back of the book says they are linearly independent. Where went wrong?

A similar example is $$\phi_1(x)=x^2, \phi_2(x)=x|x|, x\in(-\infty,\infty).$$The answer is also linear independence, but I think they are linear dependence.

Thanks for your answer.

Best Answer

Note: I wonder what DEQ this would be a solution for? Regardless, lets proceed.

We know that if $f(x)$ and $g(x)$ are linearly dependent on $I$ then $W(f,g)(x) = 0$ for all $x$ in the interval $I$.

Is this telling you anything about the linear dependence of the functions themselves? It does not imply that if $W(f,g)(x) = 0$ then $f(x)$ and $g(x)$ are linearly dependent. It is possible for two linearly independent functions to have a zero Wronskian!

We would analyze

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

  • $x < 0 \rightarrow |x| = -x \rightarrow \text{Wronskian} = 0$ since $fg' - gf' = x(-1) - (-x)(1) = 0$
  • $x = 0 \rightarrow |x| = 0 \rightarrow \text{Wronskian} = 0$ since $fg' - gf' = 0-0 = 0$
  • $x > 0 \rightarrow |x| = x \rightarrow \text{Wronskian} = 0$ since $fg' - gf' = x(1) - (x)(1) = 0$

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:

  • $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x \lt 0$. Thus, our equations to check for linear independence of these functions become:

$$c_1 x + c_2 x = 0~~~~ \text{for}~ x \ge 0 \\ c_1x - c_2 x = 0~~~~\text{for}~ x \lt 0$$

The only solution to this system is $c_1 = c_2 = 0 \rightarrow$ linear independence. Note that at the single point $x = 0$ does not matter.

You can also see the same argument for your second example Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$?

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