The Wronskian is defined by
for given functions and .
Let and be two solutions of the second-order linear differential equation
such that is not constant.
- Let be any solution of the given differential equation. Use the properties of the Wronskian proved in the previous two exercises (here and here) to prove that there exist constants and such that
- Prove that every solution of the differential equation has the form .
- Proof. By the previous exercise, since is not constant we know . So we may define,
This implies
and
- Proof. Let be the Wronskian of and . If is any solution then the Wronskian of any pair of is a solution of . Hence,
This implies,
and
Since is constant we know for any . Hence, . Therefore, by part (a),