(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
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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
-
-
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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