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I am trying to reduce the following ODE to Bessel's ODE form and hence solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$ I tried to solve it via the standard method, i.e.,...

I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...

I am trying to find the inverse of the following kernel in 3 dimensions $$ \nabla^2-x^2, $$ where, $$ x^2=\vec{x}.\vec{x} $$ It seems quit simple and one would think there should already be solutions ...

I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from: http://...

How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with $$ \big(xJ_a'(\ell x) \big)'+\left(\...

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...