Determining whether two functions are linear independent via wronskian

Homework Statement

Determine which of the following pairs of functions are linearly independent.

(a) $f(t)=3t,g(t)=|t|$

(b) $f(x)=x^{2},g(x)=4|x|^{2}$

Homework 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

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

Dick
Homework Helper
|x|^2 is equal to x^2.

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?

Dick
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.
The specific place where your calculations are incorrect is$$\frac d {dx}|x|^2 = 2|x|$$is false.