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# Use the Wronskian to find all solutions of a given second-order differential equation

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.

1. 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 2. Prove that every solution of the differential equation has the form .

1. Proof. By the previous exercise, since is not constant we know . So we may define, This implies and 2. 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), # Prove the Wronskian satisfies a first-order differential equation

The Wronskian is defined by for given functions and .

Let be the Wronskian of two solutions and of the differential equations where and are constants.

1. Prove that satisfies the first-order linear differential equation and hence, By this formula we can see that if then for all .

2. Assume is not identically zero and prove that if and only if is constant.

1. First, we evaluate where is the Wronskian of the two functions and . Furthermore, by Theorem 8.3 (page 310 of Apostol), since is a solution of we know since . Hence, if .

2. Assume . Then for all . By part (a) of the previous exercise (Section 8.14, Exercise #21) we know is constant.

Conversely, assume is constant. Then, again by the previous exericse, we have for all . Hence, # Prove some properties of the Wronskian

The Wronskian is defined by for given functions and .

1. 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 .
2. Prove that the derivative of the Wronskian is given by 1. 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 2. Proof. This is a direct computation of the derivative of the Wronskian, 