# Wronskian of $x|x|$ and $x^2$.

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.

• $x \mapsto x \vert x \vert$ is differentiable on $\mathbb R$. – 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.
• 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 $c=-1$. – Martin Argerami Feb 3 at 22:26