Example 3.  Use Frobenius series to solve the D. E.   
        [Graphics:Images/FrobeniusSeriesMod_gr_219.gif].  
Solution 3.

Determine the nature of the singularity at [Graphics:../Images/FrobeniusSeriesMod_gr_220.gif].  

[Graphics:../Images/FrobeniusSeriesMod_gr_221.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_222.gif]

Construct the Indicial Equation.

[Graphics:../Images/FrobeniusSeriesMod_gr_223.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_224.gif]

 

Find the Roots of the Indicial Equation.

[Graphics:../Images/FrobeniusSeriesMod_gr_225.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_226.gif]

 

Form the first Frobenius solution corresponding to the root  [Graphics:../Images/FrobeniusSeriesMod_gr_227.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_228.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_229.gif]

Form the set of equations to solve and do it.

[Graphics:../Images/FrobeniusSeriesMod_gr_230.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_231.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_232.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_233.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_234.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_235.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_236.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_237.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_238.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_239.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_240.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_241.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_242.gif]

The first Frobenius solution is:

[Graphics:../Images/FrobeniusSeriesMod_gr_243.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_244.gif]

Form the second Frobenius solution corresponding to the root  [Graphics:../Images/FrobeniusSeriesMod_gr_245.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_246.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_247.gif]

Form the set of equations to solve and do it.

[Graphics:../Images/FrobeniusSeriesMod_gr_248.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_249.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_250.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_251.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_252.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_253.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_254.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_255.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_256.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_257.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_258.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_259.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_260.gif]

The second Frobenius solution is:

[Graphics:../Images/FrobeniusSeriesMod_gr_261.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_262.gif]

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.

[Graphics:../Images/FrobeniusSeriesMod_gr_263.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_264.gif]

 

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_265.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_266.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_267.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_268.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_269.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_270.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_271.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_272.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_273.gif]

 

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

 

[Graphics:../Images/FrobeniusSeriesMod_gr_274.gif]

Now look at each series individually.  The first Frobenius series corresponding to  [Graphics:../Images/FrobeniusSeriesMod_gr_275.gif]  is:

[Graphics:../Images/FrobeniusSeriesMod_gr_276.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_277.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_278.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_279.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_280.gif]

Now look at the second Frobenius series corresponds to  [Graphics:../Images/FrobeniusSeriesMod_gr_281.gif].

[Graphics:../Images/FrobeniusSeriesMod_gr_282.gif]


[Graphics:../Images/FrobeniusSeriesMod_gr_283.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_284.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_285.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_286.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_287.gif]

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  [Graphics:../Images/FrobeniusSeriesMod_gr_288.gif]  instead of the recursive formula for  [Graphics:../Images/FrobeniusSeriesMod_gr_289.gif].
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.

[Graphics:../Images/FrobeniusSeriesMod_gr_290.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_291.gif]

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

[Graphics:../Images/FrobeniusSeriesMod_gr_292.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_293.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_294.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_295.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_296.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_297.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_298.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_299.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_300.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_301.gif]

We now see the pattern and define the coefficient    [Graphics:../Images/FrobeniusSeriesMod_gr_303.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_304.gif]

Compare the coefficients  [Graphics:../Images/FrobeniusSeriesMod_gr_305.gif]  with  [Graphics:../Images/FrobeniusSeriesMod_gr_306.gif].  

 

[Graphics:../Images/FrobeniusSeriesMod_gr_307.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_308.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_309.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_310.gif]

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

 

[Graphics:../Images/FrobeniusSeriesMod_gr_311.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_312.gif]
[Graphics:../Images/FrobeniusSeriesMod_gr_313.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_314.gif]

Can Mathematica find the sum of the series ?

 

[Graphics:../Images/FrobeniusSeriesMod_gr_315.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_316.gif]

Although Mathematica cannot sum the series [Graphics:../Images/FrobeniusSeriesMod_gr_317.gif],
we have at least found the formula for the coefficients.

 

[Graphics:../Images/FrobeniusSeriesMod_gr_318.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_319.gif]

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

[Graphics:../Images/FrobeniusSeriesMod_gr_320.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_321.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_322.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_323.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_324.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_325.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_326.gif]

Can Mathematica find the sum of the series ?

 

[Graphics:../Images/FrobeniusSeriesMod_gr_327.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_328.gif]

Although Mathematica cannot sum the series  [Graphics:../Images/FrobeniusSeriesMod_gr_329.gif],
we have at least found the formula for the coefficients.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004