Need help interpreting the Wronskian

[kostoglotov]

I'm given bases for a solution space \left \{ x,xe^x,x^2e^x \right \}. Clearly these form a basis (are linearly independent).

But, unless I've made a mistake, doing the Wronskian on this yields W(x) = x^3e^x.

Isn't this Wronskian equal to zero at x = 0? Isn't that a problem for dependence/independence?

note: a DE to which this solution space applies has not been provided in the exercise.

 

[Mark44]

I'm given bases for a solution space \left \{ x,xe^x,x^2e^x \right \}. Clearly these form a basis (are linearly independent).

But, unless I've made a mistake, doing the Wronskian on this yields W(x) = x^3e^x.

Isn't this Wronskian equal to zero at x = 0? Isn't that a problem for dependence/independence?

No. The three functions are linearly independent on any interval that doesn't include 0. I'm assuming that you correctly evaluated the Wronskian.