Please show work thanks :) 3. BCLet z(t) z(t) be the solution ofz" + a(t)z' + b(t)z = 0,z "+a(t)z'+b(t)z= 0, where a(t) a(t) andb(t) b(t) are continuous functions on I. Showthat y(t) = u(t)z(t) y(t)=u(t)z(t) solvesthe ODE y" + a(t)y' + b(t)y= f(t)y"+a(t)y'+b(t)y=f(t) – for some function f- when u(t) "(1) is a solution ofzu" + (2z' + az)u' = f.zu"+(2 z'+az)u'= f .

Answered: 3. BCLet z(t) z(f) be the solution of…

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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Please show work thanks :)

3. BCLet z(t) z(t) be the solution of
z" + a(t)z' + b(t)z = 0,
z "+a(t)z'+b(t)z= 0, where a(t) a(t) and
b(t) b(t) are continuous functions on I. Show
that y(t) = u(t)z(t) y(t)=u(t)z(t) solves
the ODE y" + a(t)y' + b(t)y= f(t)
y"+a(t)y'+b(t)y=f(t) – for some function f
- when u(t) "(1) is a solution of
zu" + (2z' + az)u' = f.
zu"+(2 z'+az)u'= f .

Expert Solution

Step 1

$G i v e n, z'' + a (t) z' + b (t) z = 0 . . . (1) W h e r e a (t) a n d b (t) a r e c o n t i n u o u s f u n c t i o n s o n I . T o p r o v e t h a t y (t) = u (t) z (t) s o l v e s y'' + a (t) y' + b (t) y = f (t) f o r s o m e f u n c t i o n f (t), w h e n u (t) i s a s o l u t i o n o f z u'' + (2 z' + a z) u' = f .$