Definition (Indicial Equation).  The parameter [Graphics:Images/FrobeniusSeriesMod_gr_38.gif] in the Frobenius series is a root of the indicial equation

        [Graphics:Images/FrobeniusSeriesMod_gr_39.gif].

Assuming that the singular point is  [Graphics:Images/FrobeniusSeriesMod_gr_40.gif], we can calculate [Graphics:Images/FrobeniusSeriesMod_gr_41.gif] as follows:

        [Graphics:Images/FrobeniusSeriesMod_gr_42.gif]
and
        [Graphics:Images/FrobeniusSeriesMod_gr_43.gif]

Derivation.

Starting with the differential equation  

        [Graphics:../Images/FrobeniusSeriesMod_gr_44.gif].

Rewrite it in the form  

        [Graphics:../Images/FrobeniusSeriesMod_gr_45.gif].  

Multiply each term by the factor  [Graphics:../Images/FrobeniusSeriesMod_gr_46.gif].   

        [Graphics:../Images/FrobeniusSeriesMod_gr_47.gif].  

Regroup the second and third terms as follows.  

        [Graphics:../Images/FrobeniusSeriesMod_gr_48.gif].  

Use series for all the terms including  [Graphics:../Images/FrobeniusSeriesMod_gr_49.gif].  

        [Graphics:../Images/FrobeniusSeriesMod_gr_50.gif]
        
        [Graphics:../Images/FrobeniusSeriesMod_gr_51.gif]
        
        [Graphics:../Images/FrobeniusSeriesMod_gr_52.gif]
        
        [Graphics:../Images/FrobeniusSeriesMod_gr_53.gif]

Making the substitutions we get  

    [Graphics:../Images/FrobeniusSeriesMod_gr_54.gif].  

Move the terms [Graphics:../Images/FrobeniusSeriesMod_gr_55.gif] and [Graphics:../Images/FrobeniusSeriesMod_gr_56.gif]  into the summations where they belong  

    [Graphics:../Images/FrobeniusSeriesMod_gr_57.gif].  

Now look at the first term in each of the series, multiply and add as indicated.  Mathematica can be of assistance for this computation by changing the upper limit in the summations from  [Graphics:../Images/FrobeniusSeriesMod_gr_58.gif] to [Graphics:../Images/FrobeniusSeriesMod_gr_59.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_60.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_61.gif]

This equation is  [Graphics:../Images/FrobeniusSeriesMod_gr_62.gif]  and we can cancel the common factor  [Graphics:../Images/FrobeniusSeriesMod_gr_63.gif]  and get

        [Graphics:../Images/FrobeniusSeriesMod_gr_64.gif].
        
We have arrived at the indicial equation which was our goal.  We are done!  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004