The solutions of:Originally Posted byjenjen

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:

So:

and hence:

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

RonL