(a) To determine All the regular singular points of the given differential equation. Explanation Result used: “ The regular singular points: Consider the general equation P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = 0 where x 0 is a singular point. This means that P ( x 0 ) = 0 and that at least one of Q and R is not zero at x 0 . If the value of lim x → x 0 ( x − x 0 ) Q ( x ) P ( x ) and lim x → x 0 ( x − x 0 ) 2 R ( x ) P ( x ) are finite, then the singular point x 0 is regular, otherwise it is irregular”. Calculation: The given differential equation is: x ( x − 1 ) y ″ + 6 x 2 y ′ + 3 y = 0 Compare the given differential equation with general solution of P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = 0 . Then the value P ( x ) = x ( x − 1 ) , Q ( x ) = 6 x 2 , and R ( x ) = 3 . For the singular point of the differential equation, substitute P ( x ) = 0 . P ( x ) = 0 x ( x − 1 ) = 0 x = 0 , 1 Now determine whether the singular point is regular or irregular. So, substitute x = 0 in lim x → x 0 ( x − x 0 ) Q ( x ) P ( x ) . lim x → x 0 ( x − x 0 ) Q ( x ) P ( x ) = lim x → 0 ( x − 0 ) 6 x 2 x ( x − 1 ) = lim x → 0 6 x x − 1 = 6 × 0 0 − 1 = 0 Now substitute x = 0 in lim x → x 0 ( x − x 0 ) 2 R ( x ) P ( x ) (b) To determine The indicial equation and the exponents at the singularity for each regular singular point.

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5.6, Problem 3P

(a)

To determine

All the regular singular points of the given differential equation.

(b)

To determine

The indicial equation and the exponents at the singularity for each regular singular point.

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Chapter 5.6, Problem 2P

Chapter 5.6, Problem 4P

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Elementary Differential Equations

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