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Consider the variable coefficient linear homogeneous ODE a(x)y" + b(x)y' + c(x)y = 0 where sin(z) (z² +2) sin(x)x – 2 cos(x) c(x) = 22 (cos(x)x + sin(æ)) a(x) = x-1, b(x) 2² (cos(x)x+sin(x)) A solution of the equation is y1 = sin(x). A second linearly independent solution can be found using reduction of order Y2 = UY1, where u(x) is an unknown function. The solution method involves solving a first order ODE for w(x) which determines the unknown function u() by solving another first order ODE w(x) = u'(x). Which of the following is the expression for w? sin(=) (-2 +2) - 2 cos(x) sin(x) w z(cos(x)x+sin(x)) sin(z) (z² +2) cos(x) +2 sin(x) w ¤(cos(x)x+sin(x)) sin(2)(z² +2) +2 cos(x) sin(x) w æ(cos(x)x+sin(x)) sin(x) a(=) (2²+2) cos(x) sin(x) w æ(cos(x)x+sin(x))

Consider the variable coefficient linear homogeneous ODE a(x)y" + b(x)y' + c(x)y = 0 where sin(z) (z² +2) sin(x)x – 2 cos(x) c(x) = 22 (cos(x)x + sin(æ)) a(x) = x-1, b(x) 2² (cos(x)x+sin(x)) A solution of the equation is y1 = sin(x). A second linearly independent solution can be found using reduction of order Y2 = UY1, where u(x) is an unknown function. The solution method involves solving a first order ODE for w(x) which determines the unknown function u() by solving another first order ODE w(x) = u'(x). Which of the following is the expression for w? sin(=) (-2 +2) - 2 cos(x) sin(x) w z(cos(x)x+sin(x)) sin(z) (z² +2) cos(x) +2 sin(x) w ¤(cos(x)x+sin(x)) sin(2)(z² +2) +2 cos(x) sin(x) w æ(cos(x)x+sin(x)) sin(x) a(=) (2²+2) cos(x) sin(x) w æ(cos(x)x+sin(x))

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the variable coefficient linear homogeneous ODE a(æ)y" +b(x)y' + c(x)y = 0 where sin(x) (z² +2) sin(x)x – 2 cos(x) a(x) = x1, b(æ) x² (cos(x)x+sin(x)) c(x) = x² (cos(x)x+sin(x)) A solution of the equation is y1 = sin(x). A second linearly independent solution can be found using reduction of order Y2 = uY1, where u(x) is an unknown function. The solution method involves solving a first order ODE for w(x) which determines the unknown function u(x) by solving another first order ODE w(x) = u' (x). Which of the following is the expression for w? sin(æ) (w² +2) - 2 cos(x) sin(x) w æ(cos(x)x+sin(x)) sin(z)(2² +2) cos(x) +2 sin(x) w x(cos(x)x+sin(x)) sin(x) (2² +2) +2 cos(x) sin(æ) x(cos(x)x+sin(x)) sin(2) (z² +2) cos(x) -2 a(cos(x)x+sin(x)) sin(x)
Transcribed Image Text:Consider the variable coefficient linear homogeneous ODE a(æ)y" +b(x)y' + c(x)y = 0 where sin(x) (z² +2) sin(x)x – 2 cos(x) a(x) = x1, b(æ) x² (cos(x)x+sin(x)) c(x) = x² (cos(x)x+sin(x)) A solution of the equation is y1 = sin(x). A second linearly independent solution can be found using reduction of order Y2 = uY1, where u(x) is an unknown function. The solution method involves solving a first order ODE for w(x) which determines the unknown function u(x) by solving another first order ODE w(x) = u' (x). Which of the following is the expression for w? sin(æ) (w² +2) - 2 cos(x) sin(x) w æ(cos(x)x+sin(x)) sin(z)(2² +2) cos(x) +2 sin(x) w x(cos(x)x+sin(x)) sin(x) (2² +2) +2 cos(x) sin(æ) x(cos(x)x+sin(x)) sin(2) (z² +2) cos(x) -2 a(cos(x)x+sin(x)) sin(x)
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