# Series Solutions Near a Regular Singular Point

• Mar 17th 2009, 04:45 PM
Jake8054
Series Solutions Near a Regular Singular Point
I need to find the series solution at the point x=0 for the equation 2xy''+y'+xy=0. I have already found the point to be regular singular and the roots to be 0 and 1/2. I need to solve the equation using the root 1/2.
• Mar 18th 2009, 07:10 AM
chisigma
Let’s write the equation in the form $\dots$

$y'' + \frac {y'}{2x} + \frac {y}{2}=0$ (1)

That is an linear DE and its general solution is in the form $\dots$

$y(x) = c_{1}$ $\cdot \lambda (x) + c_{2}\cdot \gamma (x)$ (2)

$\dots$ where $\lambda(*)$ and $\gamma(*)$ are two independently linear solutions of the (1) and $c_{1}$ $,c_{2}$ two constants. Now we will research a solution of (1) analytic in $x=0$, so that il can be written as $\dots$

$\lambda (x) = \sum_{n=0}^{\infty} a_{n}\cdot x^{n}$(3)

The way for finding the $a_{n}$ is to substitute the (3) into the (1) obtaining $\dots$

$\sum_{n=0}^{\infty} a_{n}\cdot x^{n}= -2\cdot \sum_{n=2}^{\infty} n\cdot (n-1)\cdot a_{n}\cdot x^{n-2} - \sum_{n=1}^{\infty} n\cdot a_{n}\cdot x^{n-2}$ (4)

The term $a_{0}$ is pratically the constant $c_{1}$ in (2). Observing the (4) it is easy to see that is $a_{1}=0$ and the same holds for all the $a_{n}$ of odd index. For the $a_{n}$ of even index we have $\dots$

$a_{0}= -2\cdot 3\cdot a_{2} \rightarrow a_{2}=-\frac {a_{0}}{2\cdot 3}$

$a_{2}= -2\cdot 2\cdot 7\cdot a_{4} \rightarrow a_{4}=-\frac {a_{2}}{2\cdot 2\cdot 7}$

$a_{4}= -2\cdot 3\cdot 11\cdot a_{6} \rightarrow a_{6}=-\frac {a_{4}}{2\cdot 3\cdot 11}$

$\dots$

$a_{2k}= -2\cdot (k+1)\cdot (4k+3)\cdot a_{2(k+1)} \rightarrow a_{2(k+1)}=-\frac {a_{2k}}{2\cdot (k+1)\cdot (4k+3)}$ (5)

The recursive relations (5) permit us to arrive to the Taylor expansion around $x=0$ of the analytic solutions of (1) $\dots$

$\lambda (x)= 1+\sum_{k=1}^{\infty} (-1)^ {k}\cdot \frac {x^{2k}}{2^{k}\cdot k!\cdot 3\cdot 7\dots (4k-1)}$ (6)

That is about the analytic solutions of the form $c_{1}\cdot \lambda(x)$. For the non analytic solutions of the form $c_{2}\cdot \gamma (x)$ the problem is a little more difficult…

Kind regards

$\chi$ $\sigma$
• Mar 20th 2009, 05:48 AM
chisigma
In previous post it has been ‘attacked’ the DE $\dots$

$y'' + \frac {y'}{2x} + \frac {y}{2}=0$ (1)

$\dots$ the general solution of which is $\dots$

$y(x) = c_{1}$ $\cdot \lambda (x) + c_{2}\cdot \gamma (x)$ (2)

For the ‘analytic’ solution $\lambda(*)$ the following series expansion has been found $\dots$

$\lambda (x) = 1+\sum_{k=1}^{\infty} (-1)^ {k}\cdot \frac {x^{2k}}{2^{k}\cdot k!\cdot 3\cdot 7\dots (4k-1)}$ (3)

… and now we will try to arrive to the ‘non analytic’ solution $\gamma (*)$. Since $\lambda (*)$ and $\gamma (*)$ are both solutions of (1) it is $\dots$

$\lambda '' + \frac {\lambda '}{2x} + \frac {\lambda}{2}=0$

$\gamma '' + \frac {\gamma '}{2x} + \frac {\gamma}{2}=0$

Multiplying the first equation by $\gamma (*)$, the second by $\lambda (*)$ and making the difference we obtain $\dots$

$\lambda '' \cdot \gamma - \gamma '' \cdot \lambda + \frac {\lambda ' \cdot \gamma - \gamma ' \cdot \lambda}{2x}=0$ (4)

$\dots$ and setting $\dots$

$\phi (x)= \lambda ' \cdot \gamma - \gamma ' \cdot \lambda$ (5)

$\dots$ the (4) becomes $\dots$

$\phi ' + \frac {\phi}{2x}=0$ (6)

The (5) is a linear DE of first order whose solution is relatively easy to find $\dots$

$\phi (x)= \frac {c}{2x}$ (7)

Substituting the (7) into (5) and setting $c=2$ without losing anything we obtain $\dots$

$x \cdot \lambda ' \cdot \gamma - x \cdot \lambda \cdot \gamma '=1$ (8)

In a succesive post we will perform succesive steps...

Kind regards

$\chi$ $\sigma$