sin 8₂ = ਸ 1²+1 Show that at this stationary value f(r) has a minimum. sin 8₁ = Exercise 1.21 Show that the functionals La S₁[y] = dr (1+ry') y' and S₂[y] = d-r √(d-x)² + h where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve, namely In(x/a) In(b/a) y(x) = A + (B-A): Explain why the da r 38 Solution 1.21 Observe that -Sa Si[y] = S₂[y] + how ?? Come dry (r) = S₂[y]+B-A, 8 = 2€ = 26 [* dx xy' (a)g'(x) + 0(e²) Chap that is, the values of the two functionals differ by a constant, independent of the path. Hence the stationary paths of the two functionals are the same. Consider the difference & Saly+eg] - S2[y], where g(a) = g(b) = 0: so that 8 = O(²) if ry'(x) = e, where e is a constant. Integrating this equation gives y(x) = d+ cln(x/a), where d is another constant. The boundary condition
sin 8₂ = ਸ 1²+1 Show that at this stationary value f(r) has a minimum. sin 8₁ = Exercise 1.21 Show that the functionals La S₁[y] = dr (1+ry') y' and S₂[y] = d-r √(d-x)² + h where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve, namely In(x/a) In(b/a) y(x) = A + (B-A): Explain why the da r 38 Solution 1.21 Observe that -Sa Si[y] = S₂[y] + how ?? Come dry (r) = S₂[y]+B-A, 8 = 2€ = 26 [* dx xy' (a)g'(x) + 0(e²) Chap that is, the values of the two functionals differ by a constant, independent of the path. Hence the stationary paths of the two functionals are the same. Consider the difference & Saly+eg] - S2[y], where g(a) = g(b) = 0: so that 8 = O(²) if ry'(x) = e, where e is a constant. Integrating this equation gives y(x) = d+ cln(x/a), where d is another constant. The boundary condition
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%
How come about I remarked on the question?
![154%
d-r
√²+h²
√(d-x)² + h
Show that at this stationary value f(x) has a minimum.
sin ₁ =
Exercise 1.21
Show that the functionals
- S dr
S₁ [y] =
sin 8₂ =
-f. dr.
dr (1+ry') y' and S₂]y] =
dx xy¹²,
Exit
where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve,
namely
y(x) = A + (B - A)
In(x/a)
In(b/a)
Explain why the same function makes both functionals stationary.
132%
38
Solution 1.21
Observe that
Si[y] = S₂[y] +
how
how me ??
+ fºdrz
dx y(x) = S₂[y] + B - A,
8 = 2€
that is, the values of the two functionals differ by a constant, independent of the
path. Hence the stationary paths of the two functionals are the same.
Consider the difference & S₂y + eg] - S2[y], where g(a) = g(b) = 0:
26 ["* dx xy' (x)g'(x) +0(2²)
Chapter I
so that 8 = O(²) if xy'(x) = c, where c is a constant. Integrating this equation
gives y(x) = d+cln(x/a), where d is another constant. The boundary conditions
now give](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2Faa2dbc98-8c58-4789-a2df-711eb4af6a42%2F0wuf6bp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:154%
d-r
√²+h²
√(d-x)² + h
Show that at this stationary value f(x) has a minimum.
sin ₁ =
Exercise 1.21
Show that the functionals
- S dr
S₁ [y] =
sin 8₂ =
-f. dr.
dr (1+ry') y' and S₂]y] =
dx xy¹²,
Exit
where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve,
namely
y(x) = A + (B - A)
In(x/a)
In(b/a)
Explain why the same function makes both functionals stationary.
132%
38
Solution 1.21
Observe that
Si[y] = S₂[y] +
how
how me ??
+ fºdrz
dx y(x) = S₂[y] + B - A,
8 = 2€
that is, the values of the two functionals differ by a constant, independent of the
path. Hence the stationary paths of the two functionals are the same.
Consider the difference & S₂y + eg] - S2[y], where g(a) = g(b) = 0:
26 ["* dx xy' (x)g'(x) +0(2²)
Chapter I
so that 8 = O(²) if xy'(x) = c, where c is a constant. Integrating this equation
gives y(x) = d+cln(x/a), where d is another constant. The boundary conditions
now give
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 2 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,
