Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Consider the Cauchy problem for the linear one-dimensional wave equation 9 Uzz for x €R and t > 0, = "n u(r,0) = f(x) for r E R, u(r, 0) = g(x) for r E R, where f e C²(R) and g E C'(R). Show that if ƒ is an odd function and g is an even function, then for every fixed t > 0, we have uz(0,t) = f'(3t).

Consider the Cauchy problem for the linear one-dimensional wave equation 9 Uzz for x €R and t > 0, = "n u(r,0) = f(x) for r E R, u(r, 0) = g(x) for r E R, where f e C²(R) and g E C'(R). Show that if ƒ is an odd function and g is an even function, then for every fixed t > 0, we have uz(0,t) = f'(3t).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

4

(a) Consider the Cauchy problem for the linear one-dimensional wave equation 9 urr for x ER and t > 0, Utt u(x, 0) = f(x) for x E R, u(x, 0) = g(x) for x E R, where f e C²(IR) and g E C'(R). Show that if f is an odd function and g is an even function, then for every fixed t > 0, we have u(0, t) = f'(3t). (b) Without proving, write down the Laplace equation in polar coordinates. Using the method of separation of variables, find a function u(r, 0) harmonic in the annulus {2 <r < 4, 0 <o < 2n} satisfying the boundary condition и(2,0) — 0, и(4,0) — sin 0.
Transcribed Image Text:(a) Consider the Cauchy problem for the linear one-dimensional wave equation 9 urr for x ER and t > 0, Utt u(x, 0) = f(x) for x E R, u(x, 0) = g(x) for x E R, where f e C²(IR) and g E C'(R). Show that if f is an odd function and g is an even function, then for every fixed t > 0, we have u(0, t) = f'(3t). (b) Without proving, write down the Laplace equation in polar coordinates. Using the method of separation of variables, find a function u(r, 0) harmonic in the annulus {2 <r < 4, 0 <o < 2n} satisfying the boundary condition и(2,0) — 0, и(4,0) — sin 0.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS