Let 0 < a < b. Consider the functional rb S[y] = [°* dx x³ (y^² – ¾y³). x5 - d²y 5 dy dx² x dx Euler-Lagrange equation for S[y] may be written as + Y1 = + y² = 0. first-integral of S[y] is 4x³yy' + x6 (y^² + ²y³) = c, where c is constant, stating any theorems that you use. Let A and B be constants. By direct substitution or otherwise, show that the functions ?? Ax-2 and y2 = B B (x² + 3/₁1) 24 both give solutions of the first-integral for any A and B, but that only one of y₁ and y2 satisfies the Euler-Lagrange equation for arbitrary A or B.
Let 0 < a < b. Consider the functional rb S[y] = [°* dx x³ (y^² – ¾y³). x5 - d²y 5 dy dx² x dx Euler-Lagrange equation for S[y] may be written as + Y1 = + y² = 0. first-integral of S[y] is 4x³yy' + x6 (y^² + ²y³) = c, where c is constant, stating any theorems that you use. Let A and B be constants. By direct substitution or otherwise, show that the functions ?? Ax-2 and y2 = B B (x² + 3/₁1) 24 both give solutions of the first-integral for any A and B, but that only one of y₁ and y2 satisfies the Euler-Lagrange equation for arbitrary A or B.
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
![Let 0 < a < b. Consider the functional
S[s] = " dar 2³ (1²³ - 3₁³).
dx x5
d²y 5 dy
dx²
x dx
Euler-Lagrange equation for S[y] may be written as
+ y² = 0.
+
Y1 =
first-integral of S[y] is
4x5 yy' + x6 (y'² + ¾y³) = c,
where c is constant, stating any theorems that you use.
Let A and B be constants. By direct substitution or otherwise, show
that the functions
??
Ax-2 and Y2
B
2
(x² + B) ²
24
both give solutions of the first-integral for any A and B, but that only
one of y₁ and y2 satisfies the Euler-Lagrange equation for arbitrary
A or B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59d44c96-efb1-4f3c-83b3-5a6a84cf94cb%2F5f0a324c-9c46-419d-bc8a-99409bcff4ad%2Fds7wk5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let 0 < a < b. Consider the functional
S[s] = " dar 2³ (1²³ - 3₁³).
dx x5
d²y 5 dy
dx²
x dx
Euler-Lagrange equation for S[y] may be written as
+ y² = 0.
+
Y1 =
first-integral of S[y] is
4x5 yy' + x6 (y'² + ¾y³) = c,
where c is constant, stating any theorems that you use.
Let A and B be constants. By direct substitution or otherwise, show
that the functions
??
Ax-2 and Y2
B
2
(x² + B) ²
24
both give solutions of the first-integral for any A and B, but that only
one of y₁ and y2 satisfies the Euler-Lagrange equation for arbitrary
A or B.
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