Partial Differentials Notes_Part_10

Partial Differentials Notes_Part_10 - two different...

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two different Frobenius expansions. Usually, this expectation is valid, but there is an important exception, which occurs when the indices differ by an integer. The general result is summarized in the following list: ( i ) If r 2 r 1 is not an integer, then there are two linearly independent solutions hatwide u ( x ) and tildewide u ( x ), each having convergent normalized Frobenius expansions of the form (12.85). ( ii ) If r 1 = r 2 , then there is only one solution hatwide u ( x ) with a normalized Frobenius expansion (12.85). One can construct a second independent solution of the form tildewide u ( x ) = log( x x 0 ) hatwide u ( x ) + v ( x ) , where v ( x ) = summationdisplay n =1 v n ( x x 0 ) n + r 2 (12 . 87) is a convergent Frobenius series that has the same index r 2 = r 1 . ( iii ) Finally, if r 2 = r 1 + k , where k > 0 is a positive integer, then there is a nonzero solution hatwide u ( x ) with a convergent Frobenius expansion corresponding to the smaller index r 1 . The second independent solution tildewide u ( x ) either has a Frobenius series expansion (12.85) with exponent r = r 2 , or an expansion of the logarithmic form (12.87). Thus, in every case, the differential equation has at least one nonzero solution with a convergent Frobenius expansion. If the second independent solution does not have a Frobenius expansion, then it requires an additional logarithmic term of a well-prescribed form. Rather than try to develop the general theory in any more detail here, we will content ourselves with the consideration of a couple of particular examples. Example 12.7. Consider the second order ordinary differential equation d 2 u dx 2 + parenleftbigg 1 x + x 2 parenrightbigg du dx + u = 0 . (12 . 88) We look for series solutions based at x = 0. Note that, upon multiplying by
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