The Wronskian is defined by
for given functions and .
- If the Wronskian of two functions and is zero for all in an open interval , prove that is constant for all . Equivalently, if is not constant on then there is some such that .
- Prove that the derivative of the Wronskian is given by
- Proof. (Note: I think we need the additional assumption that for any .) With our additional assumption we have,
since by assumption. Thus, is constant by the zero derivative theorem
- Proof. This is a direct computation of the derivative of the Wronskian,