Find the polynomials P in R[X]

[geoffrey159]

Homework Statement

Find the polynomials P in R[X] such that

$X(X+1) P'' +(X+2) P' -P = 0$​

Homework Equations

If P is a solution, I assume $P = \sum_{n=0}^N a_nX^n$

The Attempt at a Solution

I get the answer $\{\alpha (X+2),\alpha\in\mathbb{R}\}$, but I've done many calculation errors before getting it right.
Is there a way to solve that problem without too much calculation ?

 

[Mark44]

Homework Statement

Find the polynomials P in R[X] such that

$X(X+1) P'' +(X+2) P' -P = 0$​

Homework Equations

If P is a solution, I assume $P = \sum_{n=0}^N a_nX^n$

The Attempt at a Solution

I get the answer $\{\alpha (X+2),\alpha\in\mathbb{R}\}$, but I've done many calculation errors before getting it right.
Is there a way to solve that problem without too much calculation ?

When you find a solution to a differential equation, the first thing you should do is to check your solution. Have you don't this?

BTW, what does the notation R[X] mean?

 

[RUber]

Your answer does seem to work, but what is needed for higher order polynomials?

You might find it useful to use the form:
$P = \sum_{n=0}^N a_nX^n,\\ P' = \sum_{n=0}^{N-1} (n+1)a_{n+1}X^{n},\\ P'' = \sum_{n=0}^{N-2} (n+2)(n+1)a_{n+2}X^{n}$

 

[geoffrey159]

Yes I checked that it works, after the calculations, I get something like (minus the errors :-))

$0 = 2a_1 - a_0 + (N^2 -1) a_N X^N + \sum_{n=1}^{N-1} ( (n^2-1)a_n + (n+1)(n+2) a_{n+1}) X^n$.

So if P has degree N, then N = 1 and $2a_1 = a_0$.

(R[X] : Polynomials of a single variable with real coefs)

 

[RUber]

So, to do this quickly depends on your definition of quick.
$X(X+1) P'' +(X+2) P' -P = 0$

$P = \sum_{n=0}^N a_nX^n,\\ P' = \sum_{n=0}^{N-1} (n+1)a_{n+1}X^{n},\\ P'' = \sum_{n=0}^{N-2} (n+2)(n+1)a_{n+2}X^{n}$
Breaking apart the components:
$X^2 P'' +XP''+X P' +2P' - P = 0$
Gives:
$\sum_{n=0}^{N-2} (n+2)(n+1)a_{n+2}X^{n+2}\\ + \sum_{n=0}^{N-2} (n+2)(n+1)a_{n+2}X^{n+1}\\ + \sum_{n=0}^{N-1} (n+1)a_{n+1}X^{n+1}\\ +\sum_{n=0}^{N-1} 2(n+1)a_{n+1}X^{n}\\ -\sum_{n=0}^N a_nX^n =0$
Then, group all the terms by powers of X:
$X^N (N^2-N+N-1)a_N =0$
for n = 2 to N-1: $X^n [a_n(n^2-n+n-1) + a_{n+1}(n^2+n+2n+2)] = 0$
$X(2a_2+4a_2 +a_1-a_1)=0$
$1(2a_1-a_0 ) = 0$
Then it should be easy to see that for $N>1,a_N =0$ in order to push the highest order term to zero.
Quickly, you will see the solution collapse down to what you found.

I think that anytime you are playing around with indices on sums, it is easy to make mistakes...but I don't think it takes a tremendous amount of time or skill to gather exponents.

In fact, you could have just used the expansion for the highest order term to tell you that you could not have a solution of order 2 or greater, then the challenging stuff in the middle would be unnecessary.

 

[geoffrey159]

In fact, you could have just used the expansion for the highest order term to tell you that you could not have a solution of order 2 or greater, then the challenging stuff in the middle would be unecessary.

Yes, that's a shortcut I could have used. And it is way more elegant. Thank you !