Tagged Questions
2
votes
1answer
34 views
Hermite's equation of order $\alpha$
Show that the general solution of Hermite's equation of order $\alpha$:
$${y}''-2x{y}'+2\alpha y=0$$
$$is$$
$$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$
where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
0
votes
0answers
34 views
Solve For a Generating Function
Special Functions: Legendre, Bessel, Elliptic, and Confluent Hypergeometric Functions
Considering the following functions of $f_{n}(x)$ defined by:
$$
(a)---- ...
0
votes
1answer
63 views
Legendre Polynomial as Infinite Series
We've been covering Special Functions such as Legendre Functions, Bessel Functions, and Confluent Hypergeometric Functions
For: $$
f(x)=\left\{\begin{matrix} +1
& 0<x<1\\
-1 & ...
4
votes
2answers
77 views
Help with special function differential equation
this is my first time to use this site. Please let me know if the equations are unreadable, latex isn't my first language.
We've been covering Legendre, Bessel, and Confluent Hypergeometric ...
5
votes
1answer
134 views
Tough Inverse Fourier Transform
In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
0
votes
2answers
95 views
Bessel function recursion relation
I'm reading a paper and the following set of radial equations is derived:
$
-i \lbrack \partial_r + \frac{1}{r} \left( \frac{1}{2} - \nu \right) \rbrack u(r) = \pm k v(r)
$
$
-i \lbrack \partial_r ...
2
votes
1answer
128 views
Show that Bessel function $J_n(x)$ satisfies Bessel's differential equation.
here is the question:
For each positive integer $n$, the Bessel function $J_n(x)$ may be defined by
$$J_n(x) = \frac{x^n}{1\cdot 3\cdot 5\cdots(2n-1)\pi}\int^1_{-1}(1-t^2)^{n-1/2}\cos(xt) \, dt$$
...
4
votes
1answer
121 views
Euler's infinite product for the sine function and differential equation relation
Euler's infinite product for the sine function
$$\displaystyle \sin( x) = x \prod_{k=1}^\infty \left( 1 - \frac{x^2}{\pi^2k^2} \right)$$
http://en.wikipedia.org/wiki/Basel_problem
We know that ...
0
votes
2answers
47 views
Bessel function confirmation
I'm trying to see that $J_0(x)$ is indeed a solution for the Bessel equation $x^2y''+xy'+x^2y=0$, so:
$$J_0(x)=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(k!)^22^{2k}}$$
Pluging it in the equation and and ...
5
votes
0answers
96 views
Hints/Help studying an Abel Differential Equation
I want to know more than qualitative information about the Abel differential equation
$\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$
Since I don´t know how to solve this and as far as could see, this ...
0
votes
1answer
80 views
How to solve $(m_{(t)} x')' + kx = 0$ Sturm Liouville equation with bessel functions
I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to $x_{(t)}$ over the interval $t = [0, \infty]$:
$$(m_{(t)} x')' + kx = 0$$
...
0
votes
0answers
106 views
About the definition of Bessel functions of the second kind
Why Bessel functions of the second kind does not define from the second linearly independent solution of the Bessel equation that solved by Frobenius method?
For example about the Bessel function of ...
2
votes
2answers
49 views
$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$ and $\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$ where $h(x)\neq g(x)$
Is there any other solution to :
$$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$$
$$\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$$
other than $h(x)=g(x)=e^x$?
By varying $\alpha,\beta$ in
$$\frac{\mathrm{d} ...
2
votes
1answer
74 views
Show that the series representation of the Bessel function works
For the following series representation of the Bessel function:
$$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$
I want to show that w is indeed the Bessel function, such ...
2
votes
1answer
174 views
solution of Lagrange differential equation are square integrable
I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
4
votes
0answers
114 views
Solving inhomogenous bessel equation
I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
3
votes
1answer
320 views
Relationship between Legendre polynomials and Legendre functions of the second kind
I'm taking an ODE course at the moment, and my instructor gave us the following problem:
Derive the following formula for Legendre functions $Q_n(x)$ of the second kind:
$$Q_n(x) = P_n(x) ...
2
votes
1answer
75 views
To solve $U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0 $
$$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$$
$$\frac{\partial}{\partial t }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$$
...
1
vote
2answers
174 views
How do I show the Wronskian of $(J_{a}(x),Y_{a}(x)) = \dfrac {2} {\pi x}$
Based of using my undergrad class notes. I know that the wronskian of $(J_{a}(x),Y_{a}(x))$ is
$ W(J_{a}(x),Y_{a}(x)) = \left|
\begin{matrix}
J_{a}(x) & Y_{a}(x) \\
J_{a}'(x) & ...
0
votes
1answer
51 views
How to find $\lim_{x \rightarrow -N} J_{a} = (-1)^N J_{N}$?
So far I have $$\lim_{a \rightarrow -N} J_{a} = \lim_{a \rightarrow -N} \mid \frac{x}{2}
\mid^a \sum_{k=0}^{\infty} \frac{(-x^2/4)^k}{k! \; \Gamma(a+k+1)} = \mid \frac{x}{2}
\mid^{-N} ...
1
vote
1answer
149 views
To find closed form of $f(x)=\int_0^{\frac{\pi}{2}} e^{\sqrt{1-x^2 \sin^2 t}}\, dt$ as known functions
$$f(x)=\int_0^{\frac{\pi}{2}} e^{\sqrt{1-x^2 \sin^2 t}}\, dt$$
$u=\sin t$
$$f(x)=\int_0^{1} \cfrac{e^{\sqrt{1-x^2 u^2}}}{\sqrt{1-u^2}}\, du$$
$$f'(x)=\int_0^{1} \frac{-xu^2}{\sqrt{1-x^2 u^2 ...
1
vote
2answers
474 views
Problems regarding integrals involving Legendre polynomials
I am finding difficulty doing this integral involving Legendre polynomials.
$$\int_{-1}^1 x^2 P_{n-1}(x)P_{n+1}(x)dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}$$ I have two strategies in my mind both of ...
1
vote
1answer
353 views
Proving a property of Legendre polynomials containing its derivatives
I am trying to prove the following property of Legendre polynomials.
$$nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$$
My guess is that I somehow have to use the Bonnets recursion formula
...
1
vote
0answers
55 views
Functional equation for the given function
For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$
And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$.
At the same time, there is no known functional ...
2
votes
1answer
96 views
How can I express such function as known functions or power series?
$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$
$$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$
...
4
votes
1answer
96 views
Oscillation frequencies in an ODE
Given the following ODE:
$$\ddot{x}(t)+\sin(\omega t)x(t)=0$$
its solution can be expressed in terms of the Mathieu functions.
Plotting this solutions and assuming known the initial conditions it can ...
5
votes
2answers
169 views
Deriving the addition formula for the lemniscate functions from a total differential equation
The lemniscate of Bernoulli $C$ is a plane curve defined as follows.
Let $a > 0$ be a real number.
Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$.
Let $C = \{P \in ...
3
votes
1answer
73 views
Help Understanding Spectral Method for solving Differential Equations
I've posted a more detailed version of this question here : SE-ComputationalSci
but I'm really struggling with a simpler and related question. Lets say one wants to solve (I made this equation up, ...
2
votes
1answer
187 views
To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$
$3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$ where $\alpha=e^{\frac{2\pi i}{3} }$
I would like to find a closed form of $ f^{-1}(x)$
$f(x)=\sum \limits_{k=0}^\infty \frac{x^{3k}}{(3k)!}$
We can see ...
1
vote
0answers
114 views
About the Legendre differential equation
Consider the Legendre differential equation $$ (1-x^2) y'' - 2xy' + n(n+1)y = 0 $$ Then its solution is given by $$ y = c_1 P_n (x) + \text{an infinite series} $$ In fact $y = c_1 P_n (x) + c_2 Q_n ...
4
votes
1answer
239 views
To express $f(x,z)=\sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{n!}$ as known functions
$\alpha,\beta >0$
$$f(x,z)=\sum \limits_{n=0}^\infty \frac{e^{-\alpha n^2 x+\beta n z}}{n!}$$
$$\frac{\partial{f(x,z)}}{\partial z}=\beta \sum \limits_{n=1}^\infty \frac{e^{-\alpha n^2 x+\beta n ...
1
vote
0answers
108 views
How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?
How can I derive the inverse z-transform of:
$$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$
If Maple is not the way, how to derive manually?
With Maple code I encounter some problems
...
1
vote
0answers
80 views
What is the correct differential equation for the Laguerre function?
I would like to derive the correct Laguerre function from the differential equation
but the differential equations seems different from the original one.
What is the correct differential equation and ...
9
votes
4answers
547 views
Solving $(t^2+1)(y''-2y+1)=e^t$ with the initial conditions: $y(0)=y'(0)=1$
Since it is important to me I would like to award a user who would kindly explain me what are my mistakes and what is the correct way to solve the whole problem with 500 points.
I'd really like your ...
4
votes
2answers
201 views
To find closed form of $\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt $
Let $x\geq 0$, then
$$\int_0^{\frac{\pi}{2}} e^{-x\tan t+\alpha t} \;dt = U_{\alpha} (x) $$
$$-\int_0^{\frac{\pi}{2}} \tan t \ e^{-x\tan t+\alpha t} \;dt = \frac{d (U_{\alpha} (x) )}{dx} $$
...
2
votes
1answer
112 views
To simplify $f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$
Let $x\leq0$, then $$ f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$$
$$ f'(x)= -\int_{-a}^{+a} \frac{1}{t^2-a^2} e^ {-\frac{x}{t^2-a^2}}dt$$
$$ f'(x)= -\int_{-a}^{+a} ...
8
votes
3answers
257 views
Nicer expression for the following differential operator
I have the following sequence of differential operators:
$$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$
Is there any expression involving a sum of "normal" ...
2
votes
0answers
46 views
Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
2
votes
0answers
117 views
Hermite functions and integral
Let
$$
h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2},
$$
where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function.
Consider
$$
...
1
vote
0answers
54 views
$n$-th derivative of the prolate spheroidal function
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
0
votes
1answer
104 views
How to determinate the linearly independence between some special functions defined by ODE?
How to determinate the linearly independence between some special functions defined by ODE? For example:
${}_1F_1(a;b;x)$ , $x^{1-b}{}_1F_1(a-b+1;2-b;x)$ when $b$ is integer
${}_2F_1(a,b;c;x)$ , ...
0
votes
0answers
78 views
question on the expansion of the function
For a given real number $c>0$ define functions $\left(\psi_k^c(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
6
votes
2answers
166 views
Power series $x f''(x) + f'(x) + xf(x) = 0$
Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$.
How should ...
0
votes
0answers
220 views
Help with Legendre polynomials and Bessel functions…
Does anyone know of a good website to understand how to use these in solving differential equations? I'm in a class now, and I just can't seem to wrap my mind around it.
3
votes
0answers
100 views
Satisfying a Differential Equation and complex Laguerre
I have the following problem
Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
6
votes
1answer
210 views
$f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$?
$$f(x)=\int_0^{+\infty} e^{-(t+\frac{1}{t})x}\;dt$$
if while $ x>0 $ , $ f(x) $ has values
I noticed some interesting relations for $f(x)$ as shown below:
$$
\begin{align}
t & ...
4
votes
1answer
139 views
Using the Lambert W to express a solution of a differential equation.
I solved a differential equation some time ago and I need to solve for $y$. How can we solve for $y$ using the Lambert W function?
$$C_1+x = e^y+Cy$$
2
votes
1answer
191 views
Solving Shallow water Equations with Hermite polynomials
I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line.
The main equation is
$$\frac{d^2\eta}{dt^2} + ...
4
votes
3answers
286 views
Theory of the Mathieu Operator
How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it?
The Mathieu operator is an ordinary periodic ...
12
votes
1answer
482 views
Recursive solutions to linear ODE.
When finding the solutions to the simple ODE
$$ y'- mxy= x^n \text{ ; } y(0) = 0$$
I found the following:
Let $P_n$ be the particular solution for each integer exponent $n$. Then if we define
...
