a) After using the ansatz u(x, t) = X(z)T(t), which equations do you get? Pick only two. (below C' denotes a constant) a²T(t) a²X(z) a³X(z) √3CT(t) 0 a³T(t) 2² = 1 T(t) √C X(1) 0 a³X(z) 8₂² = C X(T) Əz² a³r(t) 2² 3CT(t) = = — X(z)
a) After using the ansatz u(x, t) = X(z)T(t), which equations do you get? Pick only two. (below C' denotes a constant) a²T(t) a²X(z) a³X(z) √3CT(t) 0 a³T(t) 2² = 1 T(t) √C X(1) 0 a³X(z) 8₂² = C X(T) Əz² a³r(t) 2² 3CT(t) = = — X(z)
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
![c)
Given the solution for X(x), what is the solution for T(t)?
o T(t) = e
3n²x²t
○ T(t) = b₁ cos(3²¹nt) + b2 sin(3²¹_¹nt) O T(t) = b₁ cos(3nnt) + b₂ sin(3nnt)
O T(t) = b₁ cos(√3nnt) + b₂ sin(√3nπt)
d)
[D/HD] A different PDE has a full general solution is of the form
3n²²t
u(x,t) = Σ sin(n × ππ)xe
n=1
which of the following is a possible solution consistent with the initial conditions
u(x, t0) =
= sin(87x) + sin(157x)
O u(2,t)= sin(15 × #T) xe
3x225m²t
Ⓒu(x, t) = sin(8 × πx) × e+³×64²t + sin(15 × T) × e+3x225x²t
Ou(x, t) sin(8 × π) × е
-3x64m²t
Ⓒu(x, t) = sin(8 × π™) × е
3x225m²t
+ sin(15 xx) x e
-3x64x²t
○ u(x, t) = sin(8 × π) × (cos(√3×8 × t) +5 sin(√3 x 8 x πt))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53ce1657-b18a-4e5a-99aa-3e87bdade9e3%2F9c62714c-a791-4982-a43a-a8b9aa404768%2Fehok0js_processed.png&w=3840&q=75)
Transcribed Image Text:c)
Given the solution for X(x), what is the solution for T(t)?
o T(t) = e
3n²x²t
○ T(t) = b₁ cos(3²¹nt) + b2 sin(3²¹_¹nt) O T(t) = b₁ cos(3nnt) + b₂ sin(3nnt)
O T(t) = b₁ cos(√3nnt) + b₂ sin(√3nπt)
d)
[D/HD] A different PDE has a full general solution is of the form
3n²²t
u(x,t) = Σ sin(n × ππ)xe
n=1
which of the following is a possible solution consistent with the initial conditions
u(x, t0) =
= sin(87x) + sin(157x)
O u(2,t)= sin(15 × #T) xe
3x225m²t
Ⓒu(x, t) = sin(8 × πx) × e+³×64²t + sin(15 × T) × e+3x225x²t
Ou(x, t) sin(8 × π) × е
-3x64m²t
Ⓒu(x, t) = sin(8 × π™) × е
3x225m²t
+ sin(15 xx) x e
-3x64x²t
○ u(x, t) = sin(8 × π) × (cos(√3×8 × t) +5 sin(√3 x 8 x πt))
Transcribed Image Text:Let us solve the following equation via separation of variables
J²u(x, t)
Ət²
= 3.
a²u(x, t)
?х2
a)
After using the ansatz u(x, t) = X(x)T(t), which equations do you get? Pick only two. (below C' denotes a constant)
Ə²X(1)
J²X(z)
a²T(t)
842
√3CT(t)
a²T(t)
84²
= 1 T(t)
8²X(z)
Əx²
√C X(T)
= C X(T)
8x²
Ər²
a²T(t)
3C T(t)
b)
We are interested in the oscillatory solution given by
X(x) = a₁ cos (
$(√|C|x) + a2 sin (√√C\x)
If the boundary conditions are
u(x = 0,t) = 0
u(x = 1,t) = 0
Identify the correct form of the solution:
O X(x) = a₂ sin(n+x) O X(x) = a₂ cos(NTT)
O X(z) = a2 sin((2n-1))
=
=
= X(T)
=
OX(x) = a₂ cos((2n-1) -x)
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 2 steps with 1 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,
