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.
- Prove that
satisfies the first-order linear differential equation
and hence,
By this formula we can see that if
then
for all
.
- Assume
is not identically zero and prove that
if and only if
is constant.
- 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
.
- 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,
In part b), why it is said that if W(0) = 0, then W(x) = 0 for all x?