1. ## Series Solutions Near an Ordinary Point help please

xy'' + y' + xy=0 at ordinary point x=1.

2. With the substitution $\displaystyle x-1=\xi$ the equation becomes...

$\displaystyle (1+\xi)\cdot y^{''} + y^{'} + (1+\xi)\cdot y=0$ (1)

That is an 'incomplete' linear ODE and we will search an analytic solution written as...

$\displaystyle y(\xi)= \sum_{n=0}^{\infty} a_{n}\cdot \xi^{n}$ (2)

If we derive from (2) le derivatives of $\displaystyle y(\xi)$ and subsitute them in (1) we arrive to write the following 'infinite sysytem' of algebric equations...

$\displaystyle a_{n-3} + a_{n-2} + n\cdot (n-1)\cdot (a_{n-1} + a_{n})=0$ (3)

... whose solution is...

$\displaystyle a_{n}= -a_{n-1} - \frac{ a_{n-2} + a_{n-3}}{n\cdot (n-1)}$ (4)

Since a solution of (1) multiplied by a constant is also a solution of (1), we can set without limitations $\displaystyle a_{0}=1$ and so with (4) we derive...

$\displaystyle a_{1}= -1, a_{2}= \frac{1}{2}, a_{3}= -\frac{1}{2}, a_{4}= \frac{13}{24}, a_{5}= - \frac{13}{24}, ...$

The searched solution od (1) is then...

$\displaystyle y(x)= 1 - (x-1) + \frac{1}{2}\cdot (x-1)^{2} - \frac{1}{2}\cdot (x-1)^{3} + \frac{13}{24}\cdot (x-1)^{4} - \frac{13}{24}\cdot (x-1)^{5} + ...$ (5)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$