(a) the stationary paths of the functional | S[y] = [d dx F(x,y,y', y″), y(a) = A, y(b) = B, satisfy the Euler-Lagrange equation මෙ d² OF dx2 ay" ;) d 'ƏF` dx dy' მყ OF + = 0, y(a)=A,_y(b) = B, Fy"|q = Fy" |₁ = 0. (b) Apply the result found in part (a) to the functional defined in equation (10.1) (page 219), with p = constant and the boundary conditions y(0) = y(L) = 0, to derive the associated Euler-Lagrange equation and show that its solution is y(x) = pg -x (L − x) (L² + xL — x²) . 24k - -
(a) the stationary paths of the functional | S[y] = [d dx F(x,y,y', y″), y(a) = A, y(b) = B, satisfy the Euler-Lagrange equation මෙ d² OF dx2 ay" ;) d 'ƏF` dx dy' მყ OF + = 0, y(a)=A,_y(b) = B, Fy"|q = Fy" |₁ = 0. (b) Apply the result found in part (a) to the functional defined in equation (10.1) (page 219), with p = constant and the boundary conditions y(0) = y(L) = 0, to derive the associated Euler-Lagrange equation and show that its solution is y(x) = pg -x (L − x) (L² + xL — x²) . 24k - -
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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Question
Can you show part b how to solve and more details
![(a)
the stationary paths of the functional
| S[y] = [d
dx F(x,y,y', y″), y(a) = A, y(b) = B,
satisfy the Euler-Lagrange equation
මෙ
d² OF
dx2
ay"
;)
d 'ƏF`
dx dy' მყ
OF
+
=
0, y(a)=A,_y(b) = B, Fy"|q = Fy" |₁ = 0.
(b) Apply the result found in part (a) to the functional defined in equation (10.1)
(page 219), with p = constant and the boundary conditions y(0) = y(L) = 0, to
derive the associated Euler-Lagrange equation and show that its solution is
y(x) =
pg
-x (L − x) (L² + xL — x²) .
24k
-
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F25a87098-854f-463f-b70b-1f80165e2aad%2Fjxz02j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a)
the stationary paths of the functional
| S[y] = [d
dx F(x,y,y', y″), y(a) = A, y(b) = B,
satisfy the Euler-Lagrange equation
මෙ
d² OF
dx2
ay"
;)
d 'ƏF`
dx dy' მყ
OF
+
=
0, y(a)=A,_y(b) = B, Fy"|q = Fy" |₁ = 0.
(b) Apply the result found in part (a) to the functional defined in equation (10.1)
(page 219), with p = constant and the boundary conditions y(0) = y(L) = 0, to
derive the associated Euler-Lagrange equation and show that its solution is
y(x) =
pg
-x (L − x) (L² + xL — x²) .
24k
-
-
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