This linear ODE has characteristic equation
with roots
, which gives solutions of the form
There are three cases to consider:
(1) If
, then the solution will be exactly what we see above. However, the initial conditions force both
.
(2) If
, we're left with
where
is added to the solution set to account for a second solution that is linearly independent of the first solution. Again, we get
.
(3) If
, then the square root introduces a factor of
that admits the solution
In this case, we arrive at
, and from the second condition we get
In order that
, we require that
, where
is any integer. Solving for
, we get
When
, we arrive at
, but remember that we're assuming that
, so logically the three smallest values of
that are allowed occur for
. (
, so we can just look at positive integers
.)
Unfortunately, I'm not sure exactly what's going on next. Checking with a computer, the solution is supposed to be
(Again, not sure why this is the case, but let's move on.) When
, we have the least values, which are, respectively,