# Tagged Questions

19 views

### Eigenvalue of Heun's function and its computation

It is known that the Heun's differential equation: \frac{d^2 w}{dz^2} + (\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a})\frac{dw}{dz}+\frac{\alpha \beta z -q}{z(z-1)(z-a)} ...
49 views

### Solution of $\frac{dx}{dt}=-\frac{(\sigma+1)x}{\sigma x+1}$ in terms of Lambert $w$ function

Solution of $\frac{dx}{dt}=-\frac{(\sigma+1)x}{\sigma x+1}$ in terms of Lambert $w$ function. Should I first take the solution of ODE and then apply Laplace transform. Please give step by step ...
32 views

### Solution of $\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$

How do we solve $$\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$$ I suspect it will be a function of many cases. The solution of $$\Pi(x+1)+\sin(x)=y(x)+y'(x)$$ is hard only at the evaluation of the last integral ...
45 views

329 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 ...
43 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? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
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 ...
38 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 ...
83 views

67 views

148 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 ...
67 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 ...
166 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 ...
We've been covering Special Functions such as Legendre Functions, Bessel Functions, and Confluent Hypergeometric Functions For: $$f(x)=\left\{\begin{matrix} +1 & 0<x<1\\ -1 & ... 2answers 166 views ### Help with special function differential equation this is my first time to use this site. Please let me know if the equations are unreadable, latex isn't my first language. We've been covering Legendre, Bessel, and Confluent Hypergeometric ... 1answer 417 views ### Tough Inverse Fourier Transform In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ... 1answer 506 views ### Show that Bessel function J_n(x) satisfies Bessel's differential equation. here is the question: For each positive integer n, the Bessel function J_n(x) may be defined by$$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$$... 1answer 379 views ### Euler's infinite product for the sine function and differential equation relation Euler's infinite product for the sine function$$\displaystyle \sin( x) = x \prod_{k=1}^\infty \left( 1 - \frac{x^2}{\pi^2k^2} \right)$$http://en.wikipedia.org/wiki/Basel_problem We know that ... 2answers 144 views ### Bessel function confirmation I'm trying to see that J_0(x) is indeed a solution for the Bessel equation x^2y''+xy'+x^2y=0, so:$$J_0(x)=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(k!)^22^{2k}}$$Pluging it in the equation and and ... 0answers 185 views ### Hints/Help studying an Abel Differential Equation I want to know more than qualitative information about the Abel differential equation \frac{dy}{dx}+y^3+x=0. \qquad ... \;(1) Since I don´t know how to solve this and as far as could see, this ... 1answer 190 views ### How to solve (m_{(t)} x')' + kx = 0 Sturm Liouville equation with bessel functions I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to x_{(t)} over the interval t = [0, \infty]:$$(m_{(t)} x')' + kx = 0$$... 2answers 51 views ### \frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x) and \frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x) where h(x)\neq g(x) Is there any other solution to :$$\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} ...
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 ...