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Solve the differential equation y"-9y=0 using the power series method. Example problem and  solutions in the picture.

Solve the differential equation y"-9y=0 using the power series method. Example problem and  solutions in the picture.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Solve the differential equation y"-9y=0 using the power series method. Example problem and  solutions in the picture.
54 olution: Assuming a solution in the form y = Co + C,x+ C,x2 + C,x' + C.x + Cx + CoX ** ++ 2C,x + 3C,x2 + 4C,x + 5C,x + 6C6X + y" = 2C, + 6C,x + 12C,x2 + 20C,x + 30C6*" * Hence, y' e y" + y = 0 (2C, + 6C3X + 12C,x2 + 20C,x + 30Cx + ...) + (Co + C,x + C2x2 + C,x + Cax* + ...) = 0 Collecting like powers of x. (2C2 + Ca) + (6C, + C,)x + (12C, + C,)x? + (20C, + C,)x + (30C% + C,)x* = 0 Then, a) 2C2 + Co = 0 -Ca C, = - b) 6C + C, = 0 C, = %3D c) 12C, + C, = 0 Co C2 C = - 12 12 c. Co ;, C = 24 d) 20C, + C, = 0 C; = 20 C = 20 120
Transcribed Image Text:54 olution: Assuming a solution in the form y = Co + C,x+ C,x2 + C,x' + C.x + Cx + CoX ** ++ 2C,x + 3C,x2 + 4C,x + 5C,x + 6C6X + y" = 2C, + 6C,x + 12C,x2 + 20C,x + 30C6*" * Hence, y' e y" + y = 0 (2C, + 6C3X + 12C,x2 + 20C,x + 30Cx + ...) + (Co + C,x + C2x2 + C,x + Cax* + ...) = 0 Collecting like powers of x. (2C2 + Ca) + (6C, + C,)x + (12C, + C,)x? + (20C, + C,)x + (30C% + C,)x* = 0 Then, a) 2C2 + Co = 0 -Ca C, = - b) 6C + C, = 0 C, = %3D c) 12C, + C, = 0 Co C2 C = - 12 12 c. Co ;, C = 24 d) 20C, + C, = 0 C; = 20 C = 20 120
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