(a) To determine The indicial equation and its roots for the point x = 1 by using the given data. Explanation The Legendre equation of order α is ( 1 − x 2 ) y " − 2 x y ' + α ( α + 1 ) y = 0 . Add and subtract 1 in the left side of the Legendre equation as shown below: ( 1 − x 2 ) y " − 2 x y ' + α ( α + 1 ) y = [ 1 − [ ( x − 1 ) + 1 ] 2 ] y " − 2 ( ( x − 1 ) + 1 ) y ' + α ( α + 1 ) y = [ 1 − ( x − 1 ) 2 − 1 − 2 ( x − 1 ) ] y " − 2 ( ( x − 1 ) + 1 ) y ' + α ( α + 1 ) y = [ − ( x − 1 ) 2 − 2 ( x − 1 ) ] y " − 2 ( ( x − 1 ) + 1 ) y ' + α ( α + 1 ) y Further simplification of the above equation as shown below: ( 1 − x 2 ) y " − 2 x y ' + α ( α + 1 ) y = − ( x − 1 ) ( ( x − 1 ) + 2 ) y " − 2 ( ( x − 1 ) + 1 ) y ' + α ( α + 1 ) y = [ − ( x − 1 ) 2 − 2 ( x − 1 ) ] y " + [ − 2 ( x − 1 ) − 2 ] y ' + α ( α + 1 ) y Consider that the solution y is given in the form y = ∑ n = 0 ∞ a n ( x − 1 ) r + n . Differentiate the above equation y = ∑ n = 0 ∞ a n ( x − 1 ) r + n is as follows: y ' = d d x ( ∑ n = 0 ∞ a n ( x − 1 ) r + n ) = ∑ n = 0 ∞ ( r + n ) a n ( x − 1 ) r + n − 1 Again, differentiate the equation y ' = ∑ n = 0 ∞ ( r + n ) a n ( x − 1 ) r + n − 1 is as follows: y " = d d x ( ∑ n = 0 ∞ ( r + n ) a n ( x − 1 ) r + n − 1 ) = ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n ( x − 1 ) r + n − 2 Substitute the value of y , y ' and y " in the Legendre equation is as follows: [ − ( x − 1 ) 2 − 2 ( x − 1 ) ] ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n ( x − 1 ) r + n − 2 + [ − 2 ( x − 1 ) − 2 ] ∑ n = 0 ∞ ( r + n ) a n ( x − 1 ) r + n − 1 + α ( α + 1 ) ∑ n = 0 ∞ a n ( x − 1 ) r + n = 0 By multiplying the factors with the corresponding summations, the equation becomes as follows: − ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n ( x − 1 ) r + n − 2 ∑ n = 0 ∞ ( r + n ) ( r + n − 1 ) a n ( x − 1 ) r + n − 1 − 2 ∑ n = 0 ∞< (b) To determine The series solution in powers of x − 1 for x − 1 > 0 by using the given data.

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5.5, Problem 11P

(a)

To determine

The indicial equation and its roots for the point x=1 by using the given data.

(b)

To determine

The series solution in powers of x−1 for x−1>0 by using the given data.

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Chapter 5.5, Problem 10P

Chapter 5.5, Problem 12P

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