Example 2.  Use Frobenius series to solve the D. E.   
        .  
Solution 2.

Determine the nature of the singularity at .  

Construct the Indicial Equation.

Find the Roots of the Indicial Equation.

Form the first Frobenius solution corresponding to the root  .

Form the set of equations to solve and do it.

The first Frobenius solution is:

Form the second Frobenius solution corresponding to the root  .

Form the set of equations to solve and do it.

The second Frobenius solution is:

After you are done, use Mathematica's DSolve subroutine to get the answer and check out its series expansion.
This will require slight fussing around with the appropriate multiple of  "Sinh".

At this time we could plot the series approximations and the analytic solutions. To see the difference in the graphs we will reduce the number of terms in the series.

The recursive formula for the coefficients.

    If we look at the series in more depth we will be able to obtain the analytic solutions as infinite sums.  First find the recursive formula for the coefficients of  .  If you try this be sure to use the  " := "  replacement delayed structure to avoid an infinite recursion.  For this example the trial term   works with the replacements .  

If you can't get the above computation to work, then just type in the recursive formula.

Now look at each series individually.  The first Frobenius corresponds to  .  

Now look at the second Frobenius series corresponding to  .  

The explicit formulas for the coefficients for the first Frobenius series are:

Note. We cannot add up the infinite number of terms in this sequence because Mathematica does not know this formula yet.

Remark. However, Mathematica can treat infinite series "like a function."

(c) John H. Mathews 2004