dy (a) With x = z2 the chain rule gives dx Y(z) = y(2²), so the functional becomes S[Y] = 2 = 2 /² [² dz ² ( 12 (1²) ² z dz 0 rZ = = [² d= dz dy dz dy 1 dz dx dz dx/dz – w2y2 ( 1 y¹² - 12w²y²). The Euler-Lagrange equation is now d 'Y' dz Z which expands to zY" - Y' + 4z³w²Y = 0. = ) = + 4zw²Y = 0, 1 dy 2z dz where Z = X¹/2, " where How Come &
dy (a) With x = z2 the chain rule gives dx Y(z) = y(2²), so the functional becomes S[Y] = 2 = 2 /² [² dz ² ( 12 (1²) ² z dz 0 rZ = = [² d= dz dy dz dy 1 dz dx dz dx/dz – w2y2 ( 1 y¹² - 12w²y²). The Euler-Lagrange equation is now d 'Y' dz Z which expands to zY" - Y' + 4z³w²Y = 0. = ) = + 4zw²Y = 0, 1 dy 2z dz where Z = X¹/2, " where How Come &
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
Question
100%
![X
Exercise 6.1 (R) The functional S[y] = f* dx (y¹² - w²y²), where w is a constant,
gives rise to the Euler-Lagrange equation y" +w²y = 0.
(a) Show that changing the independent variable to z where x = z² gives the functional
Z = √X,
S[Y]
=
Z
1
4x
dz
(Y' (2)² – 46²zY²),
Z
with the associated Euler-Lagrange equation
d²Y
dz²
Z
dy
dz
+4w²z³Y = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2Fb254e101-2e65-483f-a630-4beeb78cbce9%2Fbjliv3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:X
Exercise 6.1 (R) The functional S[y] = f* dx (y¹² - w²y²), where w is a constant,
gives rise to the Euler-Lagrange equation y" +w²y = 0.
(a) Show that changing the independent variable to z where x = z² gives the functional
Z = √X,
S[Y]
=
Z
1
4x
dz
(Y' (2)² – 46²zY²),
Z
with the associated Euler-Lagrange equation
d²Y
dz²
Z
dy
dz
+4w²z³Y = 0.
![dy
dx
(a) With x = z2 the chain rule gives
Y(z) = y(z²), so the functional becomes
S[Y]:
=
2
Z
[² d
0
dz z
Z
1
-² [² d²
=
dz
1
42²
dy
dz
=
The Euler-Lagrange equation is now
2
dy dz
dz dx
-w²y²
( 1⁄y¹2² – 4zw²y²).
=
which expands to 2Y" - Y' + 4z³w²Y = 0.
dt (7) + 4zw²Y = 0,
dz
dy 1
dz dx/dz
=
1 dy
2z dz
where_Z = X¹/2,
where
How Come ??](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2Fb254e101-2e65-483f-a630-4beeb78cbce9%2Ftriee2q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:dy
dx
(a) With x = z2 the chain rule gives
Y(z) = y(z²), so the functional becomes
S[Y]:
=
2
Z
[² d
0
dz z
Z
1
-² [² d²
=
dz
1
42²
dy
dz
=
The Euler-Lagrange equation is now
2
dy dz
dz dx
-w²y²
( 1⁄y¹2² – 4zw²y²).
=
which expands to 2Y" - Y' + 4z³w²Y = 0.
dt (7) + 4zw²Y = 0,
dz
dy 1
dz dx/dz
=
1 dy
2z dz
where_Z = X¹/2,
where
How Come ??
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 4 steps with 3 images
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,
