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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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