Recall that the general solution of the 1-dimensional wave equation with fixed ends, that is, the following boundary value problem on the interval [0, π] Utt = Uxx # |u(0, t) = u(π, t) = 0 is given by † u(x, t) = [an sin() cos(µ) + b₂ sin(™) sin(™)]. n=0 † u(x, t) = [an sin(nx) cos(nt) + bn sin(nx) sin(nt)]. n=0 Define on [0, π] b(x) = x= [²] otherwise. a. Find the solution of # with the following initial conditions: u(x,0) = b(x), ut(x,0) = 0. b. (*) Find the solution for u(x, 0) = 0, ut(x, 0) = b(x). c. (**) Use the product to sum formulæfor sin and cos to show that any solution given in † can be written as F(x + t) + G(x − t) for two functions F and G.
Recall that the general solution of the 1-dimensional wave equation with fixed ends, that is, the following boundary value problem on the interval [0, π] Utt = Uxx # |u(0, t) = u(π, t) = 0 is given by † u(x, t) = [an sin() cos(µ) + b₂ sin(™) sin(™)]. n=0 † u(x, t) = [an sin(nx) cos(nt) + bn sin(nx) sin(nt)]. n=0 Define on [0, π] b(x) = x= [²] otherwise. a. Find the solution of # with the following initial conditions: u(x,0) = b(x), ut(x,0) = 0. b. (*) Find the solution for u(x, 0) = 0, ut(x, 0) = b(x). c. (**) Use the product to sum formulæfor sin and cos to show that any solution given in † can be written as F(x + t) + G(x − t) for two functions F and G.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
Related questions
Question
![Recall that the general solution of the 1-dimensional wave equation with fixed ends, that is, the following
boundary value problem on the interval [0, π]
Utt = Uxx
#
u(0, t) = u(π, t) = 0
is given by
†
u(x, t) = [an sin() cos(x) + b₂ sin() sin()].
bn
n=0
†
u(x, t) = [an sin(nx) cos(nt) + bn sin(nx) sin(nt)].
n=0
Define on [0, π]
b(x)
x = []
- {
=
otherwise.
a. Find the solution of # with the following initial conditions: u(x, 0) = b(x), ut(x,0) = 0.
b. (*) Find the solution for u(x, 0) = 0, ut(x,0) = b(x).
c. (**) Use the product to sum formulæfor sin and cos to show that any solution given in † can be
written as F(x + t) + G(x − t) for two functions F and G.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91f76606-a4d9-42f0-be0d-7b1366a5593f%2Fb197a73f-f215-42c2-bb73-3a798bac5ac9%2Fh4il68_processed.png&w=3840&q=75)
Transcribed Image Text:Recall that the general solution of the 1-dimensional wave equation with fixed ends, that is, the following
boundary value problem on the interval [0, π]
Utt = Uxx
#
u(0, t) = u(π, t) = 0
is given by
†
u(x, t) = [an sin() cos(x) + b₂ sin() sin()].
bn
n=0
†
u(x, t) = [an sin(nx) cos(nt) + bn sin(nx) sin(nt)].
n=0
Define on [0, π]
b(x)
x = []
- {
=
otherwise.
a. Find the solution of # with the following initial conditions: u(x, 0) = b(x), ut(x,0) = 0.
b. (*) Find the solution for u(x, 0) = 0, ut(x,0) = b(x).
c. (**) Use the product to sum formulæfor sin and cos to show that any solution given in † can be
written as F(x + t) + G(x − t) for two functions F and G.
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