Series Solution
A series solution expands when closed forms are unavailable: substituting into the ODE and equating coefficients of each power of produces a recurrence relation for the coefficients.
Suppose we want to find a series solution about to the ODE
Since is an ordinary point, we expect two linearly independent series solutions.
We shall try
So we have
and
For convenience, we shall multiply the ODE by to get
Substituting the series into the ODE gives
Equating coefficients of for gives
Hence we have a recurrence relation for . Therefore, and are arbitrary constants in the general solution.
Consider the odd terms. Note that . Hence, all odd terms are zero. Therefore, one solution is
Consider the even terms. We have
Therefore
Hence, the other solution is
Note that
Therefore, we can write the even solution as
Hence, we can write the general solution as
which is a closed form solution as well.
Note the behavior near , near the regular singular points.
Working method
To find a series solution about :
- Try and differentiate term by term.
- Multiply the ODE by a convenient power of so that every term carries the same power before substituting.
- Substitute the series and shift indices so that all sums share the same power of .
- Equate coefficients of to obtain a recurrence relation relating to earlier coefficients. The constants and stay arbitrary, giving the two arbitrary constants of the general solution.
For about , the recurrence is . The odd branch terminates since , so one solution is . The even branch gives
using , so the general solution closes in closed form. The behavior near reflects the regular singular points there.
Related
Stated in
- Example 8.5 (Series Solutions About Ordinary Point)§8.2 Method of Frobenius