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.
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.
Let’s write the equation in the form
(1)
That is an linear DE and its general solution is in the form
(2)
where and are two independently linear solutions of the (1) and two constants. Now we will research a solution of (1) analytic in , so that il can be written as
(3)
The way for finding the is to substitute the (3) into the (1) obtaining
(4)
The term is pratically the constant in (2). Observing the (4) it is easy to see that is and the same holds for all the of odd index. For the of even index we have
(5)
The recursive relations (5) permit us to arrive to the Taylor expansion around of the analytic solutions of (1)
(6)
That is about the analytic solutions of the form . For the non analytic solutions of the form the problem is a little more difficult…
Kind regards
In previous post it has been ‘attacked’ the DE
(1)
the general solution of which is
(2)
For the ‘analytic’ solution the following series expansion has been found
(3)
… and now we will try to arrive to the ‘non analytic’ solution . Since and are both solutions of (1) it is
Multiplying the first equation by , the second by and making the difference we obtain
(4)
and setting
(5)
the (4) becomes
(6)
The (5) is a linear DE of first order whose solution is relatively easy to find
(7)
Substituting the (7) into (5) and setting without losing anything we obtain
(8)
In a succesive post we will perform succesive steps...
Kind regards