example of two linearly independent functions to have a zero Wronskian??

Question

What is an example of two linearly independent functions to have a zero Wronskian??

This question is in reference to http://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx

Answer 1

$$ f(x)=x^2$$

$$g(x) = x|x|$$

Note that $$f'(x)=2x$$ while $$ g'(x)=2x$$ for $x\ge 0$

and $$ g'(x)=-2x$$ for $x\le 0$.

Thus $W(f,g)=0$ for all $x$ while $f$ and $g$ are not linearly dependent on $(-\infty, \infty)$

Answer 2

The standard example (due to Peano) are the functions $f(x)=x^2$ and $g(x)=x|x|$.

Answer 3

Let's see what $0$ Wronskian of two function means: $$W(f,g) = f g' - f'g =0$$ On any interval $I$ on which $g\ne 0$ we have $$\left(\frac{f}{g}\right)' = \frac{f'g-f g'}{g^2} = 0$$ so on that interval $\frac{f}{g}$ is constant.

So let's take $f$ to be $0$ on $(-\infty, 0]$, and $\ne 0$ on $(0, \infty)$, and $g(x) = f(-x)$. Then $W(f,g) \equiv 0$ but clearly $f$, $g$ are not proportional. For instance, $f(x) = e^{-\frac{1}{x}}$ for $x> 0$, and $0$ on $(-\infty, 0]$, $g(x) = e^{\frac{1}{x}}$ for $x<0$ and $0$ on $[0, \infty)$.