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Thus the finite solutions to the given differential equation, for any value of n , must be of the following form: 2 1/2 ( ) ( ) (1 ) ndP d Combining the elements of the solution we have, for any value of n : 2 1/2 ( , ) ( )(1 ) ( ) where cos n n nr A R r Substituting the foregoing results for ( ) and ( )n nR r into the expression for the stream function yields the result: 2 2 ( )(1 ) ( , ) sin (cos ) n n n n nn dP r A r d d A P r d This solution is valid for any value of the integer n , so ...

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