# Determining whether two functions are linear independent via wronskian

 P: 6 1. The problem statement, all variables and given/known data Determine which of the following pairs of functions are linearly independent. (a) $f(t)=3t,g(t)=|t|$ (b) $f(x)=x^{2},g(x)=4|x|^{2}$ 2. Relevant equations the Wronskian is defined as, W=Det{{f(u),g(u)},{f'(u),g'(u)}} if {f(u),g(u)} are linearly dependent, W=0 if W=/=0, {f(u),g(u)} are linearly independent 3. The attempt at a solution The assumed interval for the independent variables t,x are $x,t \in (-\infty,\infty)$ for (a), I determined $W(t)=3t-3|t|$, which for x>0 is $W(t)=3t-3t=0$. for x<0 we have $W(t)=-3t-3t=-6t$. Since for some value of $t \in (-\infty,\infty)$ I found a $W(t) \neq 0$ I can conclude that the functions f(t) and g(t) are linearly independent. Now for (b), Similarly to (a), I find a $W(x)=8x^{2}|x|-8|x|^{2}x$ for $x>0 : W(x)=8x^{3}-8x^{3}=0$ for $x<0 : W(x)=8(-x)^{2}|-x|-8|-x|^{2}(-x)=8x^{3}+8x^{3}=16x^{3}\neq 0$ Similarly I conclude that f(x) and g(x) are linearly independent since I found values of x which make the wronskian not equal to 0. However, while my conclusion is correct for (a), (b) is supposedly linearly dependent. Is my method correct? if so what mistake did I make in concluding that the functions of (b) were linearly independent? Thanks
 P: 6 Yes, I use that fact, but I still find that for $x<0 : W(x)=8(-x)^{2}|-x|-8(-x)^{3}=8x^{3}+8x^{3}=16x^{3}$ Am I just completely missing something here?
 Sci Advisor HW Helper Thanks P: 25,228 Determining whether two functions are linear independent via wronskian Your calculation of $W(x)=8x^{2}|x|-8|x|^{2}x$ is wrong. Just use that |x|^2=x^2 from the beginning, so g(x)=4x^2.
 HW Helper Thanks PF Gold P: 7,632 The specific place where your calculations are incorrect is$$\frac d {dx}|x|^2 = 2|x|$$is false.