Example 1.  Use Frobenius series to solve the D. E.   
        [Graphics:Images/FrobeniusSeriesMod_gr_68.gif].  
Solution 1.

Determine the nature of the singularity at [Graphics:../Images/FrobeniusSeriesMod_gr_69.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_70.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_71.gif]

Construct the Indicial Equation.

[Graphics:../Images/FrobeniusSeriesMod_gr_72.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_73.gif]

 

Find the Roots of the Indicial Equation.

[Graphics:../Images/FrobeniusSeriesMod_gr_74.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_75.gif]

 

Form the first Frobenius solution corresponding to the larger root  [Graphics:../Images/FrobeniusSeriesMod_gr_76.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_77.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_78.gif]

Form the set of equations to solve and do it.

[Graphics:../Images/FrobeniusSeriesMod_gr_79.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_80.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_81.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_82.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_83.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_84.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_85.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_86.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_87.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_88.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_89.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_90.gif]

The first Frobenius solution is:

[Graphics:../Images/FrobeniusSeriesMod_gr_91.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_92.gif]

Form the second Frobenius solution corresponding to the smaller root  [Graphics:../Images/FrobeniusSeriesMod_gr_93.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_94.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_95.gif]

Form the set of equations to solve and do it.  

[Graphics:../Images/FrobeniusSeriesMod_gr_96.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_97.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_98.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_99.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_100.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_101.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_102.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_103.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_104.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_105.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_106.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_107.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_108.gif]

The second Frobenius solution is:

[Graphics:../Images/FrobeniusSeriesMod_gr_109.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_110.gif]

Observe that the coefficients that involve [Graphics:../Images/FrobeniusSeriesMod_gr_111.gif] are that multiple of the first Frobenius solution.  Hence we can set  [Graphics:../Images/FrobeniusSeriesMod_gr_112.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_113.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_114.gif]

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

[Graphics:../Images/FrobeniusSeriesMod_gr_115.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_116.gif]

Now we plot the series approximations and the analytic solutions.

[Graphics:../Images/FrobeniusSeriesMod_gr_117.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_118.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_119.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_120.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_121.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_122.gif]

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  [Graphics:../Images/FrobeniusSeriesMod_gr_123.gif].  If you try this be sure to use the  " := "  replacement delayed structure to avoid an infinite recursion.  For this example the trial term  [Graphics:../Images/FrobeniusSeriesMod_gr_124.gif] works with the replacements [Graphics:../Images/FrobeniusSeriesMod_gr_125.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_126.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_127.gif]

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

 

[Graphics:../Images/FrobeniusSeriesMod_gr_128.gif]

Now look at each series individually.  The first Frobenius series corresponds to  [Graphics:../Images/FrobeniusSeriesMod_gr_129.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_130.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_131.gif]

Now look at the second Frobenius series which corresponds to  [Graphics:../Images/FrobeniusSeriesMod_gr_132.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_133.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_134.gif]

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

[Graphics:../Images/FrobeniusSeriesMod_gr_135.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_136.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_137.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_138.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_139.gif]

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

[Graphics:../Images/FrobeniusSeriesMod_gr_140.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_141.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_142.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_143.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_144.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004