Thread: A differential equation

I won't be posting new problems for a couple of weeks. This one is nice:


Suppose and is a function (see here for the definition) and for all Show that is a polynomial of degree at most

Originally Posted by NonCommAlg

I won't be posting new problems for a couple of weeks. This one is nice:


Suppose and is a function (see here for the definition) and for all Show that is a polynomial of degree at most

if we assume that f(x) is a polynomial of the degree n-1,

we get,



If f(x) has the degree n-2,

then the (n-1)th derivative of f becomes zero.

The same holds for all degrees <(or equal to) (n-1)

hence, the given product always becomes vanishes.


However, if we now start making the degrees > (n-1)

Lets assume that the function is a polynomial of degree n.

I'll take the simplest case here. assuming



Then, the nth derivative of the function:



And all the other derivatives would be of the form:




it follows that

that none of the derivatives will now be zero hence, the product cannot be zero.