2
votes
1answer
77 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
1
vote
2answers
79 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
0
votes
0answers
19 views

Constant Coefficient Legendre Equation via Change of Variables?

In the introduction to this old book by Craig on ode's it is said that The theory of linear differential equations may almost be said to find its origin in Fuchs's two memoirs published in 1866 ...
0
votes
0answers
16 views

Sturm Liouville eigenvalues eigenfunctions

The equation/Sturm Liouville problem is: $$u'' + \lambda u = 0, \quad 0≤x≤\frac{\pi}{2}, \quad u'(0) = 0, \quad u(\frac{\pi}{2}) = 0 $$ I want to find the eigenvalues and eigenfunctions and the ...
1
vote
1answer
74 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
5
votes
1answer
310 views

Generalized Legendre differential equation

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ which is ...
0
votes
1answer
39 views

Satisfaction of Bessel equation by any other function.

Is it possible that any function $y(x)$ other than Bessel group of functions, satisfy Bessel's equation? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
0
votes
0answers
22 views

Solving non-linear, but separable and autonomous, matrix ODE $H'(z) = A H(z)^k + S$

Start with the non-linear scalar ODE: $H'(z) = A H(z)^k + S$. You can separate and integrate this to find something like: $$z + C_1 = \int_0^{H}\frac{1}{S - A q^k}dq$$ From this, you can use the ...
0
votes
1answer
25 views

Reduction of DEs to Bessel equation

A question in my textbook asks me to write down the general solution to: $\frac{d}{dx}(x^2\frac{dR(x)}{dx}) + [k^2x^2 - n(n+1)]R(x) = 0$ in terms of Bessel functions. Now two similar questions ...
3
votes
1answer
81 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
0
votes
1answer
39 views

Solution to Legendre eq in trig form

Okay I'm having a little trouble in answering this question... so the general solution is $y(x) = AP_n(x) + BQ_n(x)$ umm then what do I do?
1
vote
0answers
16 views

mathieu function of non integer order asymptotics

I have an asymptotic expression for the integer mathieu functions: $se_\nu(q,z)$ and $ce_\nu(q,z)$, where $\nu$ is an integer. I would like to use these expression for the case $\nu$ real. My question ...
0
votes
0answers
28 views

eigenfunctions in a Sturm-Liouville problem

I've found that the eigenfunctions in a certain Sturm-Liouville problem satisfy a differential equation whose general solution is $\phi(x)= x^{a}[C_1M(a,2a+2,x)+C_2U(a,2a+2,x)]$, $x\ge0$, where $M$ is ...
1
vote
1answer
32 views

What will be the solution of this equation?

What will be the solution of the equation. $(x^2+m^2)\frac{\partial^2y}{\partial(x^2+m^2)}+(x+m)\frac{\partial y}{\partial (x+m)}+(x^2+m^2-n^2)=0$ where $m$ may be a constant
0
votes
1answer
39 views

ODE second order and Bessel function

I have a function $xy''-y'+y=0$ that I'm trying to solve. I thought of solving it like this $ y''-y'/x+y/x=0 $ this you can write as $(y'/x)'+y=0 $ and then you can find the solution as a Bessel ...
0
votes
0answers
42 views

Limits of generalized hypergeometric functions

For a (quite fiddly) asymptotic matching, I would like to be able to write the solution to \begin{equation} \frac{\mathrm{d}^5}{\mathrm{d}x^5}f(x) + \frac{10}{15^{1/2}} \frac{\mathrm{d}}{\mathrm{d}x} ...
1
vote
0answers
45 views

Integral Solutions to Special Function Equations Like Legendre's Equation

Using Laplace's method one is able to find solutions to ode's with scary names like Cauchy: $a_2y'' + a_1y' + a_0y = 0$ Laplace: $(a_2+b_2x)y'' + (a_1+b_1x)y' + (a_0+b_0x)y = 0$ Euler: $a_2x^2y'' ...
0
votes
2answers
207 views

Second order linear ODE, self adjoint (Sturm-Liouville) form. Orthogonality of solutions - confused about the weight factor.

If I have an ODE of the form $$a(x)y''+b(x)y'+c(x)y= \lambda y$$ Such that $b=a'$, then it is equivalent to: $$(a(x)y')'+c(x)y= \lambda y$$ So the solutions corresponding to two different ...
1
vote
0answers
73 views

Fabius function and equivalent

The Fabius function $F$ can be defined on $[0,1]$ by $F(0)=0$ $F(1)=1$ on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$ on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$ It's a known example of a not analytic ...
1
vote
1answer
100 views

Bessel function with complex argument

So I understand that the bessel functions of the first kind are the ones that satisfy this equation: $$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\alpha^2)y = 0$$ and the result is a linear ...
1
vote
1answer
100 views

How to use following equation by using Green's function?

Let's have the following equation: $$ u''(r) + \frac{1}{r}u'(r) - \alpha^{2}u(r) = f(r), $$ where $r$ is polar radius. Method of Green's function leads to $$ u''(r) + \frac{1}{r}u'(r) - ...
2
votes
1answer
67 views

Delta function proof in QM

I'm actually working with some QM problems at the moment but I've hit a wall with a delta potential involved. The problem asks me to verify that $$ \frac{d \phi_{x=0^{+}}}{dx} -\frac{d ...
2
votes
4answers
106 views

Solve $\frac{d}{dx}f(x)=f(x-1)$

I am trying to find a function such that $\dfrac{d}{dx}f(x)=f(x-1)$ Is there such function other than $0$ ?
5
votes
2answers
217 views

4th order differential equations with Bessel Function solutions

I am working on a a 4th order linear PDE coming from the modified wave equation of a stiff material. I have radial symmetry which has lead me to a 4th order ODE in $r$: $r^3 R''''(r) + 2r^2 R'''(r) - ...
0
votes
0answers
65 views

Positive, increasing function satisfying a differential equation and an asymptotic property

I'm looking for a (any) differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ with the following properties: $\forall x \in \mathbb{R}, f(x) > 0 \wedge f'(x) >0$ $\forall a \in ...
1
vote
1answer
283 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
1
vote
1answer
385 views

Eigenvalues and Eigenfunctions of a singular Sturm-Liouville operator using Bessel functions

I’m trying to find the eigenvalues and eigenvectors of the Singular Sturm-Liouville operator: $$Lu=xu''+u'$$ $$u(1)=0$$ $$u(0) \text{ is finite}$$ $$0 < x < 1$$ My approach to solving ...
3
votes
1answer
79 views

Bessel equation relation

Define Bessel function as: $$J_a(x)=\sum_{n=0}^{\infty}{{(-1)^n}\over{\Gamma(a+n+1)n!}}\left({{x}\over{2}}\right)^{a+2n}.$$ Where $a$ not an integer. Need to show ...
4
votes
1answer
60 views

Bessel function values

Given $$J_m(x)=\sum_{n=0}^{\infty}{{(-1)^n}\over{n!(n+m)!}}\left(\frac{x}{2}\right)^{m+2n},$$ where $m=0,1,2,\ldots$ and $x\ge0$. Need to show $$\left|J_m(x)\right|\le1.$$
0
votes
2answers
73 views

How to determine if $f_n(x) $ is periodic function or not for $n>2$

$f_n(x)$ is defined as $$\int_0^{f_n(x)}\! \frac{\mathbb{d}t}{1+t^n}=x$$ $$\frac{d}{dx}\int_0^{f_n(x)}\! \frac{\mathbb{d}t}{1+t^n}=\frac{d (x)}{dx}$$ $$f'_n(x) \frac{1}{1+f^n_n(x)}=1$$ $$f'_n(x) ...
2
votes
0answers
143 views

Diffusion in Spherical Coordinates with mixed BC

I have been working through the book "A Guide to First-Passage Processes" and wanted to branch out on my own doing a calculation similar to what occurs in chapter 6. My basic problem comes from the ...
4
votes
1answer
66 views

Uniqueness of weight function.

Let $L=p(x)\frac{d^2}{dx^2}+q(x)\frac{d}{dx}+r(x).$ Where L stands for differential operator. Now inner product defined $(f,g)=\int_a^bf(x)g(x)w(x)dx$. Where $w(x)$ is a weight function. Now $L$ is ...
2
votes
1answer
150 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
1answer
302 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
164 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 ...
12
votes
1answer
402 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 ...
1
vote
2answers
729 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
494 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
364 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
139 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 ...
6
votes
0answers
177 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
180 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 $t = [0, \infty]$: $$(m_{(t)} x')' + kx = 0$$ ...
2
votes
2answers
50 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
183 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 ...
5
votes
1answer
262 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
238 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
497 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
86 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
569 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
54 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} ...