The solutions of:Originally Posted by jenjen
are all of the form:
where and are arbitrary constants and
are also solutions of the DE.
Now lets try as a trial solution of the equation:
Now if the 's are real the solutions have at most a finite number
of zeros, so for there to be an infinite number of zeros of
we need the 's to be complex so then we must have (Note
this does not guarantee that the solution has any zeros for real ,
but solutions which are real will have an infinite number of zeros).
Also for to die away as we need
that , so . This is because:
is oscillatory with increasing amplitude if , of constant amplitude
if and of decaying amplitude if .
Hence for to have an infinite number of zeros and to die
away as we need