**Example 3.** Use
Frobenius series to solve the D. E.

.

**Solution 3.**

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 some fussing around with the appropriate multiple
of the Bessel function.

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. Also, include the semicolon at the end of the
lines.

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 series corresponding to is:

Now look at the second Frobenius series corresponds to .

**Note.** We cannot add up an
infinite number of terms in this sequence because we do not have a
closed formula for the coefficients c[k], it is a recursive
formula and will exceed the finite recursion depth of
*Mathematica*.

**Remark.** We would like to have
a explicit formula for instead
of the recursive formula for .

Such formulas need to be discovered. How can we do
in. For this example, we can resort to "picking"
*Mathematica*'s mind.

That is to say, we can look at the way *Mathematica* solves it
and reverse engineer the solution.

Since *Mathematica* thinks that the Gamma function is used,
we will discover "how to do it."

We now see the pattern and define the coefficient

Compare the coefficients with .

Now sum the series to get the first few terms in the solution.

Can *Mathematica* find the sum of the series ?

Although *Mathematica* cannot sum the series ,

we have at least found the formula for the coefficients.

Similarly, we can find a formula for the coefficients of the second solution.

Can *Mathematica* find the sum of the series ?

Although *Mathematica* cannot sum the
series ,

we have at least found the formula for the coefficients.

(c) John H. Mathews 2004