Determining whether two functions are linear independent via wronskian

[ysebastien]

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&gt;0 : W(x)=8x^{3}-8x^{3}=0
for x&lt;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]

|x|^2 is equal to x^2.

 

[ysebastien]

Yes,

I use that fact, but I still find that for
x&lt;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.

 

[LCKurtz]

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

 

[ysebastien]

Ah, I see. Thank you