Wikipedia says wronskian of $x|x|$ and $x^2$ is identically zero. But it is not LD.

I know why these two are LI and not LD.

since x|x| is not differentiable function,how to find their wronskian???? And plz suggest ways to check LI and LD when functions are not differentiable.

Thanks in advance. Plz help.

  • $\begingroup$ What means LI and LD? $\endgroup$ – mathcounterexamples.net Feb 3 at 19:27
  • $\begingroup$ linearly independent and linearly dependent @mathcounterexamples.net $\endgroup$ – Aryadeva Feb 3 at 19:27
  • $\begingroup$ $x \mapsto x \vert x \vert$ is differentiable on $\mathbb R$. $\endgroup$ – mathcounterexamples.net Feb 3 at 19:29

The function $f(x)=x|x|$ is is differentiable everywhere. For $x>0$, you have $f(x)=x^2$, differentiable. For $x<0$ you have $f(x)=-x^2$, differentiable. At $0$, you have $$ \frac{f(h)-f(0)}h=\frac{h|h|}h=|h|\to0, $$ so the derivative exists and is zero.

For two functions, using the Wronskian is overkill. Linear dependence for two functions means that one is a multiple of the other: this is trivial to check for yes or for no. In your example, for instance, if $x|x|=cx^2$ for all $x$, then evaluate at $1$ to get $c=1$, and at $-1$ to get $c=-1$; so such $c$ cannot exist and the functions are linearly independent.

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  • $\begingroup$ What about linear Independence of |x| and x? (Here, |x| is not differentiable at x=0) and what to do for other non-differentiable functions??? $\endgroup$ – pankaj kumar Feb 3 at 21:40
  • $\begingroup$ Exactly what I said applies: if $|x|=cx$, evaluate at any positive $x$ and you get $c=1$; evaluate at any $x<0$ and you get $c=-1$. $\endgroup$ – Martin Argerami Feb 3 at 22:26

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