To determine A lower bound for the radius of convergence of the series of solution for ( x 2 − 2 x − 3 ) y ″ + x y ′ + 4 y = 0 with the initial values x 0 = 4 , x 0 = − 4 and x 0 = 0 . Explanation Theorem used: If x 0 is an ordinary point of the differential equation P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = 0 , that is if p = Q P and q = R P are analytic at x 0 , then the general solution of differential equation is y = ∑ n = 0 ∞ a n ( x − x 0 ) n = a 0 y 1 ( x ) + a 1 y 2 ( x ) , where a 0 and a 1 are arbitrary, and y 1 and y 2 are the two power series solutions which are analytic at x 0 . The solutions y 1 and y 2 form a fundamental set of solutions. Further, the radius of convergence for each of the series solutions y 1 and y 2 is at least as large as the minimum of the radii of convergence of the series for p and q . Calculation: Compare the given equation ( x 2 − 2 x − 3 ) y ″ + x y ′ + 4 y = 0 with P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = 0 . Then, P ( x ) = x 2 − 2 x − 3 , Q ( x ) = x and R ( x ) = 4 are analytic at x 0 = 4 . Equate P ( x ) to 0 and obtain the values of x . P ( x ) = 0 x 2 − 2 x − 3 = 0 x 2 + x − 3 x − 3 = 0 x ( x + 1 ) − 3 ( x + 1 ) = 0 ( x + 1 ) ( x − 3 ) = 0 Then, the values of x are –1 and 3 which are different from the given x 0

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5.3, Problem 6P

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A lower bound for the radius of convergence of the series of solution for (x2−2x−3)y″+xy′+4y=0 with the initial values x0=4,x0=−4 and x0=0.

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Chapter 5.3, Problem 5P

Chapter 5.3, Problem 7P

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Elementary Differential Equations

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