I don't why al=(2n-1)pi/2. Can you please explain it to me
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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I don't why al=(2n-1)pi/2. Can you please explain it to me?Thank you
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Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2, 0 < x < L , t &...
| Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u dt2,0 < x < L , t > 0 ...
DIFFERENTIAL
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ndary Value Proo
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9th Edition
Dennis G. Zill
Publisher: Cengage Learning
DENNIS G ZILL
Find 2
ISBN: 9781337604901
Chapter 12.7, Problem 6E
Textbook Problem
Solve the boundary-value problem
a?
dx2
0 < x < L, t> 0
||
dt2
ди
u (0, t) = 0, Eª = Fo, t> 0
dx [x=L
ди
и (х, 0) %3D 0, 0, 0 <х < L.
|f=0
The solution u(x, t) represents the longitudinal displacement of a vibrating elastic bar that is
anchored at its left end and is subjected to a constant force of magnitude F, at its right end. See
Figure 12.4.7 in Exercises 12.4. Eis a constant called the modulus of elasticity.
Expert Solution
To determine
The longitudinal displacement u (x, t) of a vibrating elastic bar such that
a² *u = &u_u (0,1) = 0, u (x. 0) = 0, E = Fo and L = 0.
dx Ix=L
dt Ix=L
X GET 10 FREE QUESTIONS
Answer to Problem 6E"
Transcribed Image Text:10:58 АМ Тhu May 6
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AA
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M (108) - F G cascade yar...
M (38,939) - s...
y!
Yahoo
PM Trigonometri...
Solve the bo...
= bartleby
Q Search for textbooks, step-by-step explanatio...
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Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2, 0 < x < L , t &...
| Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u dt2,0 < x < L , t > 0 ...
DIFFERENTIAL
EQUATIONS wth
ndary Value Proo
Bundle: Differential Equations wit...
9th Edition
Dennis G. Zill
Publisher: Cengage Learning
DENNIS G ZILL
Find 2
ISBN: 9781337604901
Chapter 12.7, Problem 6E
Textbook Problem
Solve the boundary-value problem
a?
dx2
0 < x < L, t> 0
||
dt2
ди
u (0, t) = 0, Eª = Fo, t> 0
dx [x=L
ди
и (х, 0) %3D 0, 0, 0 <х < L.
|f=0
The solution u(x, t) represents the longitudinal displacement of a vibrating elastic bar that is
anchored at its left end and is subjected to a constant force of magnitude F, at its right end. See
Figure 12.4.7 in Exercises 12.4. Eis a constant called the modulus of elasticity.
Expert Solution
To determine
The longitudinal displacement u (x, t) of a vibrating elastic bar such that
a² *u = &u_u (0,1) = 0, u (x. 0) = 0, E = Fo and L = 0.
dx Ix=L
dt Ix=L
X GET 10 FREE QUESTIONS
Answer to Problem 6E

Transcribed Image Text:10:58 АМ Тhu May 6
* 56%
AA
bartleby.com
+
M (108) - F G cascade yar...
M (38,939) - s...
y!
Yahoo
PM Trigonometri...
Solve the bo...
= bartleby Q Search for textbooks, step-by-step explanatio...
Ask an Expert
Math / Bundle: Differential Equations w... / Solve the boundary-value problem a 2 d 2 u d x 2 = d 2 u d t 2 , 0 < x < L , t &...
: Solve the boundary-value problem a 2 d 2 u d x 2= d 2 u dt 2,0 < x < L ,t > 0 ...
X (x) = c8 sin (ax)
(11)
Differentiate the above equation with respect to x.
X' (x)
= ac6cos (ax)
Substitute L for x in the above equation.
X
X' (L) =
= ac8cos (aL)
ac8cos (aL) = 0
(x' (L) = 0)
(2n – 1) 5
(2n – 1)
aL
a =
Therefore, the Eigen values are as follows
An = an
= ((2n – 1) )
Substitute the value of a in equation (11).
X (x) = cg sin ((2n – 1)
TX
2L
The above equation represents the Eigen functions where. n = 1, 2, 3, ....
Substitute a? for 1 in equation (6).
T" + a²a?T = 0
The general solution of the above equation is as follows:
T (t) = c9cos (aat) + c10sin (aat)
(12)
Differentiate the above equation with respect to t.
T' (t)
= -aac9 sin (aat) + aac10 cos (aat)
Substitute 0 for t in the above equation.
X GET 10 FREE QUESTIONS )
(T' (0) = 0)
Expert Solution
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