**Example 1.** Use
Frobenius series to solve the D. E.

.

**Solution 1.**

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 larger root .

Form the set of equations to solve and do it.

The first Frobenius solution is:

Form the second Frobenius solution corresponding to the smaller root .

Form the set of equations to solve and do it.

The second Frobenius solution is:

Observe that the coefficients that involve are that multiple of the first Frobenius solution. Hence we can set .

After you are done, use *Mathematica*'s DSolve subroutine to
get the answer and check out its series expansion.

Now we plot the series approximations and the analytic solutions.

**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 series corresponds to .

Now look at the second Frobenius series which corresponds to .

When the explicit formulas for the coefficients of the first Frobenius are used we get:

When the explicit formulas for the coefficients of the second Frobenius are used we get:

(c) John H. Mathews 2004