Solve the following initial boundary value problem for the wave equation: Utt = 16uz, u,(0, t) = u_(4, t) = 0, u(z,0) = 0, u,(z,0) = 2z + 1, 00 (1)| t>0 (2) 0
Solve the following initial boundary value problem for the wave equation: Utt = 16uz, u,(0, t) = u_(4, t) = 0, u(z,0) = 0, u,(z,0) = 2z + 1, 00 (1)| t>0 (2) 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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![Solve the following initial boundary value problem for the wave equation:
16uzz,
u, (0, t) = u,(4, t) = 0,
u(x, 0) = 0, u(z, 0) = 2x +1,
0<z< 4, t >0 (1)|
t>0 (2)
0 <z<4 (3)
Derive an expression for all nontrivial product solutions
u(z, t) = X(z)T(t)|
of (1) satisfying boundary conditions (2), and solve the corresponding eigenvalue problem.
The eigenvalues are
,n = 1,2,...|
The corresponding eigenfunctions are
Xp(x) =
n = 1,2,...|
F(2)"X
All product solutions of equation (1) satisfying boundary conditions (2) are
uo(z, t) = } Ao + Bd|
%3D
U,(z, t) = (A||
+B
,n = 1,2,...|
where An, Bn, n = 0,1, 2, ... | are arbitrary constants.
Use Fourier series representation to find all constants
An, Bn, n = 0, 1, 2,..
such that
u(z, t) = u9(x, t) +
4„(z,t)
is a solution of problem (1)-(3).
Ag =
Bo =
A, =
n = 1,2, ...|
B =
n = 1,2, ...|](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbd1efaf2-9c0c-46e4-a460-5a6aa8f2a0ec%2F36cbe907-f1e5-4230-a746-e0d550f84395%2F9tpu1qb_processed.png&w=3840&q=75)
Transcribed Image Text:Solve the following initial boundary value problem for the wave equation:
16uzz,
u, (0, t) = u,(4, t) = 0,
u(x, 0) = 0, u(z, 0) = 2x +1,
0<z< 4, t >0 (1)|
t>0 (2)
0 <z<4 (3)
Derive an expression for all nontrivial product solutions
u(z, t) = X(z)T(t)|
of (1) satisfying boundary conditions (2), and solve the corresponding eigenvalue problem.
The eigenvalues are
,n = 1,2,...|
The corresponding eigenfunctions are
Xp(x) =
n = 1,2,...|
F(2)"X
All product solutions of equation (1) satisfying boundary conditions (2) are
uo(z, t) = } Ao + Bd|
%3D
U,(z, t) = (A||
+B
,n = 1,2,...|
where An, Bn, n = 0,1, 2, ... | are arbitrary constants.
Use Fourier series representation to find all constants
An, Bn, n = 0, 1, 2,..
such that
u(z, t) = u9(x, t) +
4„(z,t)
is a solution of problem (1)-(3).
Ag =
Bo =
A, =
n = 1,2, ...|
B =
n = 1,2, ...|
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