0
votes
1answer
72 views

First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
0
votes
2answers
67 views

Polynomial Solutions for Differential Equations

Suppose we have a set of polynomials where $\deg(Q_k(x))\le k$, and consider the following differential equation, $$W:=\sum_{k=0}^n Q_k(x)\frac{d^k}{dx^k} .$$ It is known that if there is a ...
0
votes
1answer
47 views

Rolles Theorem Simple and multiple zeros

I have this problem with Legendre polinomials Use Rolle's Theorem to show that Pn cannot have multiple zeros in the open interval (-1, 1). In other words, any zeros of Pn which lie in (-1, 1) must be ...
3
votes
1answer
90 views

Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that ...
1
vote
3answers
122 views

Differentiable function and polynomials: Proof of $\phi (x)=ce^x + q(x)$ unclear to me.

I came across a proof which I can't quite understand: If $p:\mathbb{R}\to\mathbb{R}$ is a polynomial of degree $n$ and $\phi:\mathbb{R}\to\mathbb{R}$ is a differentiable function with $\phi'=\phi+p$, ...
4
votes
1answer
116 views

Polynomial differential equation

I came across this problem in an old olympiad paper (Putnam?) Find all polynomials $p(x)$ with real coefficients satisfying the differential equation $7\dfrac{d }{dx } [xp(x)]=3p(x)+4p(x+1)$ $\ \ ...
1
vote
0answers
51 views

A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$

Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$. $F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$. $P_1(F)$ is the set of all ...
4
votes
1answer
661 views

Runge-Kutta 4 - solving system of 6 differential equations (BVP)

I'm facing a tricky problem. I need to solve a system of 6 differential equations numerically, but I don't have 6 IVP (initial value problem) conditions, instead I have 6 BVP (boundary valye problem) ...
2
votes
2answers
211 views

Prove that $f$ is differentiable (polynomial),

Let $f:\mathbb{R}^n \to \mathbb{R}$ be a polynomial in $n$ variables, i.e. $$f(x_1, \ldots, x_n) = \sum_{k_1,\ldots, k_n =0}^{m} a_{k_1\ldots k_n} x_1^{k_1} \cdots x_n^{k_n}$$ for some ...
5
votes
2answers
201 views

General solution of $C_{n+2}(x)=xC_n(x)+nC_{n-1}(x)$

Airy differential equation. $y''(x)=xy(x)$ $y'''(x)=y(x)+x y'(x)$ $y'^v(x)=x^2y(x)+2 y'(x)$ $y^v(x)=4xy(x)+x^2 y'(x)$ $y^{(6)}(x)=(x^3+4)y(x)+6x y'(x)$ . . $y^{(n)}(x)=A_n(x)y(x)+B_n(x) y'(x)$ ...
2
votes
1answer
1k views

Solving polynomial differential equation

I have $a(v)$ where $a$ is acceleration and $v$ is velocity. $a$ can be described as a polynomial of degree 3: $$a(v) = \sum\limits_{i=0}^3 p_i v^i = \sum\limits_{i=0}^3 p_i ...
12
votes
2answers
721 views

Sum of derivatives of a polynomial

Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$. Show that ...
8
votes
1answer
206 views

When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
1
vote
2answers
878 views

Show that the Hermite polynomials form a basis of $\mathbb{P}_3$

I have this question that I took a shot at but I am not very familiar with Hermite or Laguerre, this my first time running across these type of polynomials and need some help please. (a) The first ...