The differential equation dy dx? dy + y = 0 2. (4.120) dx has the general solution y(x) = (c1 + c2x)e = c1e² + c2xe*. (4.121) The associated difference equation is Yk+2 – 2yk+1+ Yk = 0. (4.122) From equation (4.121) we obtain dk dyk (Cie" + c2xe") = cje" + c2(xe“ + ke"), Dky(x) || (4.123) where the expression in parentheses on the right-hand side of equation (4.123) was obtained by using the Leibnitz rule for the kth derivative of a product. Therefore, Yk = Dky(x)|x=0 = c1 + c2k, (4.124) which is easily shown to be the general solution of equation (4.122).
The differential equation dy dx? dy + y = 0 2. (4.120) dx has the general solution y(x) = (c1 + c2x)e = c1e² + c2xe*. (4.121) The associated difference equation is Yk+2 – 2yk+1+ Yk = 0. (4.122) From equation (4.121) we obtain dk dyk (Cie" + c2xe") = cje" + c2(xe“ + ke"), Dky(x) || (4.123) where the expression in parentheses on the right-hand side of equation (4.123) was obtained by using the Leibnitz rule for the kth derivative of a product. Therefore, Yk = Dky(x)|x=0 = c1 + c2k, (4.124) which is easily shown to be the general solution of equation (4.122).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
100%
Explain the determine
Transcribed Image Text:Example B
The differential equation
dy
dy
2
+ y = 0
dx
(4.120)
dx?
has the general solution
y(x)
(C1 + c2x)e" = c1eª + c2xe“.
(4.121)
The associated difference equation is
Yk+2 – 2yk+1 + Yk = 0.
(4.122)
From equation (4.121) we obtain
Dry(x) =
dk
:(c1e + c2xe®)
dæk
(4.123)
= cje# + c2(xe® + ke“),
where the expression in parentheses on the right-hand side of equation (4.123)
was obtained by using the Leibnitz rule for the kth derivative of a product.
Therefore,
D*y(x)=0 = C1 + czk,
which is easily shown to be the general solution of equation (4.122).
Yk =
(4.124)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
