Question 4 Consider the telegraph equation Pu du - 2a - au, dt subject to the initial conditions u(x, t) = sin r, (z,0) = e', at where a is a constant. a. Show that using the transformation u(x, t) = V(r, t)e-at, then the telegraph equation becomes b. Show further that using the coordinate transformation { = 1- ct and 7 =r+ ct the wave equation in (a) above becomes = 0. Solve this equation by direct integration. c. Use the given initial conditions and confirm that the solution to the telegraph equation is u(x, t) = e-at[sin(x) cos(ct) - (a/c) sin(z) sin(ct) + sinh(ct)e"].
Question 4 Consider the telegraph equation Pu du - 2a - au, dt subject to the initial conditions u(x, t) = sin r, (z,0) = e', at where a is a constant. a. Show that using the transformation u(x, t) = V(r, t)e-at, then the telegraph equation becomes b. Show further that using the coordinate transformation { = 1- ct and 7 =r+ ct the wave equation in (a) above becomes = 0. Solve this equation by direct integration. c. Use the given initial conditions and confirm that the solution to the telegraph equation is u(x, t) = e-at[sin(x) cos(ct) - (a/c) sin(z) sin(ct) + sinh(ct)e"].
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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![Question 4
Consider the telegraph equation
Pu
du
- 2a
- au,
at
2
subject to the initial conditions
u(x, t) = sin x,
ar (x, 0) = et.
where a is a constant.
a. Show that using the transformation
u(x, t) = V(x, t)e-at,
then the telegraph equation becomes
b. Show further that using the coordinate transformation E = x - ct and
n = r+ ct the wave equation in (a) above becomes
= 0.
Solve this equation by direct integration.
c. Use the given initial conditions and confirm that the solution to the
telegraph equation is
u(x, t) = e-at [sin(x) cos(ct) - (a/c) sin(x) sin(ct) + sinh(ct)e"].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2201fdd8-619c-42d2-9691-3942ac947821%2F1ed14b25-4a35-48f4-aa18-88cb65b46e8f%2Fm3sjm6q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 4
Consider the telegraph equation
Pu
du
- 2a
- au,
at
2
subject to the initial conditions
u(x, t) = sin x,
ar (x, 0) = et.
where a is a constant.
a. Show that using the transformation
u(x, t) = V(x, t)e-at,
then the telegraph equation becomes
b. Show further that using the coordinate transformation E = x - ct and
n = r+ ct the wave equation in (a) above becomes
= 0.
Solve this equation by direct integration.
c. Use the given initial conditions and confirm that the solution to the
telegraph equation is
u(x, t) = e-at [sin(x) cos(ct) - (a/c) sin(x) sin(ct) + sinh(ct)e"].
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