# learning to solve inhomogeneous versions of Legendre, Hermite, Laguerre etc.

Does anyone know of any good links (or even books though preferably not as there's a chance I won't be able to acquire them ) which go through worked examples of solving inhomogeneous versions of the Legendre, Laguerre and Hermite equations.

Math stack exchange is great when you have a specific question , but a textbook is better when you want to learn material for the first time so I'm only really looking for external links.

To give some context here is an example of the sort of question I want to know how to solve :

Given that

$$\int_{-\infty}^{\infty}e^{-(x-a)^2}H_n(x)dx=\sqrt{\pi}2^na^n$$

Where $$H_n$$ is the Hermite polynomial of order n, with eigenvalue−2n, normalised so that $$\int^{\infty}_{\infty}H_m(x)H_n(x)e^{-x^2}dx=2^n\sqrt{\pi}n!\delta_{n,m}.$$

Find a solution to the inhomogeneous equation

$$\tfrac{d^2y}{dx^2}-2x\tfrac{dy}{dx}+y=e^{2ax}$$

a a constant.

Note: The reason I ask is I'm studying for an exam thats in a few weeks

• Hi again, did you get a chance to look at Riley, Hobson and Bence? Also, did you have a go at the extra exercise I suggested in reply to your question yesterday? Because most exam questions are special cases of that exercise. And that includes the question here! Hint: first use the orthogonality relations to write $e^{2ax}$ in the form $\sum_{n = 0}^\infty a_n H_n(x)$....] – Kenny Wong Dec 28 '18 at 22:00
• Hey :)yeah I checked out Riley,Hobson and Bence but from what I could see at a cursory glance there wasn't much in the way of examples of this kind, I haven't got a chance to work on that exercise you recommended yet (I was mainly working on solving partial differential equations through separation of variables today) I'll try to do it tomorrow though and maybe send you what I work out on the original question we spoke on,if that's okay? (I suggest the original question just because then it's a continuation of that idea so it'll be more helpful for future students if its all in the one place) – excalibirr Dec 28 '18 at 22:35
• @KennyWong wouldn't let me add an at user to my comment for some reason so I added it here just to let you know I replied :) – excalibirr Dec 28 '18 at 22:51
• Sure, good luck! – Kenny Wong Dec 28 '18 at 22:52