# Tagged Questions

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### 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 ...
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### 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? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...