Example 5.  Use Frobenius series to solve the D. E.   
        [Graphics:Images/FrobeniusSeriesMod_gr_382.gif].  
A solution is known to be the celebrated Bessel function  [Graphics:Images/FrobeniusSeriesMod_gr_383.gif].
Solution 5.

Determine the nature of the singularity at [Graphics:../Images/FrobeniusSeriesMod_gr_384.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_385.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_386.gif]

Construct the Indicial Equation.

[Graphics:../Images/FrobeniusSeriesMod_gr_387.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_388.gif]

 

Find the Roots of the Indicial Equation.

[Graphics:../Images/FrobeniusSeriesMod_gr_389.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_390.gif]

 

Form the first Frobenius solution corresponding to the root  [Graphics:../Images/FrobeniusSeriesMod_gr_391.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_392.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_393.gif]

Form the set of equations to solve and do it.

[Graphics:../Images/FrobeniusSeriesMod_gr_394.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_395.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_396.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_397.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_398.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_399.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_400.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_401.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_402.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_403.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_404.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_405.gif]

The first Frobenius solution is:

[Graphics:../Images/FrobeniusSeriesMod_gr_406.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_407.gif]

This first Frobenius series solution is a Bessel Function of the First Kind, and it is defined at the origin  [Graphics:../Images/FrobeniusSeriesMod_gr_408.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_409.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_410.gif]

The second solution to the differential equation is a Bessel Function of the Second Kind.  It is not a simple Frobenius series and will be discussed in another module.

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.

[Graphics:../Images/FrobeniusSeriesMod_gr_411.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_412.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_413.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_414.gif].  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.

[Graphics:../Images/FrobeniusSeriesMod_gr_415.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_416.gif]

 

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

 

[Graphics:../Images/FrobeniusSeriesMod_gr_417.gif]

Now look at each series individually.  The first Frobenius series corresponding to  [Graphics:../Images/FrobeniusSeriesMod_gr_418.gif]  is:

[Graphics:../Images/FrobeniusSeriesMod_gr_419.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_420.gif]

The coefficient  [Graphics:../Images/FrobeniusSeriesMod_gr_421.gif]  can be expressed in closed form    [Graphics:../Images/FrobeniusSeriesMod_gr_422.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_423.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_424.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_425.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_426.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_427.gif]

The the first Frobenius series solution is twice the Bessel function  [Graphics:../Images/FrobeniusSeriesMod_gr_428.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_429.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_430.gif]

Aside.  

The Frobenius series solution  s(x)  will have the initial conditions   s(0) = 0  will   [Graphics:../Images/FrobeniusSeriesMod_gr_431.gif], whereas the the built in Mathematica function  BesselJ[1,x]  does not have the initial value 0 because it is the negative of the derivative of  BesselJ[0,x].

[Graphics:../Images/FrobeniusSeriesMod_gr_432.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_433.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004