- #1

ysebastien

- 6

- 0

## Homework Statement

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

(a) [itex]f(t)=3t,g(t)=|t|[/itex]

(b) [itex]f(x)=x^{2},g(x)=4|x|^{2}[/itex]

## 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 [itex] x,t \in (-\infty,\infty) [/itex]

for (a),

I determined [itex] W(t)=3t-3|t| [/itex], which for x>0 is [itex] W(t)=3t-3t=0 [/itex].

for x<0 we have [itex] W(t)=-3t-3t=-6t [/itex]. Since for some value of [itex] t \in (-\infty,\infty)[/itex] I found a [itex]W(t) \neq 0[/itex] I can conclude that the functions f(t) and g(t) are linearly independent.

Now for (b),

Similarly to (a), I find a [itex] W(x)=8x^{2}|x|-8|x|^{2}x[/itex]

for [itex] x>0 : W(x)=8x^{3}-8x^{3}=0[/itex]

for [itex] x<0 : W(x)=8(-x)^{2}|-x|-8|-x|^{2}(-x)=8x^{3}+8x^{3}=16x^{3}\neq 0[/itex]

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