Explanation Given: The given heat equation is u x x = u t , 0 < x < L , t > 0 , the length of rod is L , initial temperature of the rod is u 0 + u 0 sin x π L and the temperature is kept constant at the ends x = 0 and x = L . Calculation: The given heat equation is, ∂ 2 u ∂ x 2 = ∂ u ∂ t (1) Take Laplace transform of both sides of the above equation, L { ∂ 2 u ∂ x 2 } = L { ∂ u ∂ t } d 2 U d x 2 = s U ( x , s ) − u ( x , 0 ) Substitute u ( x , 0 ) = u 0 + u 0 sin x π L in the above equation, d 2 U d x 2 − s U ( x , s ) = − u 0 − u 0 sin x π L The particular solution of the above equation is, U ( x , s ) = c 1 cosh s x + c 2 sinh s x + u 0 s + u 0 s + π 2 L 2 sin π L x �

Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term

9th Edition

ISBN: 9781337604901

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 14.2, Problem 21E

To determine

To solve: The heat equation for the temperature u(x,t).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.2, Problem 20E

Chapter 14.2, Problem 22E

Students have asked these similar questions

Please as many det

8–23. Sketching vector fields Sketch the following vector fields

25-30. Normal and tangential components For the vector field F and curve C, complete the following: a. Determine the points (if any) along the curve C at which the vector field F is tangent to C. b. Determine the points (if any) along the curve C at which the vector field F is normal to C. c. Sketch C and a few representative vectors of F on C. 25. F = (2½³, 0); c = {(x, y); y − x² = 1} 26. F = x (23 - 212) ; C = {(x, y); y = x² = 1}) , 2 27. F(x, y); C = {(x, y): x² + y² = 4} 28. F = (y, x); C = {(x, y): x² + y² = 1} 29. F = (x, y); C = 30. F = (y, x); C = {(x, y): x = 1} {(x, y): x² + y² = 1}

Chapter 14 Solutions

Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term

Ch. 14.1 - (a) Show that erf(t)=10ted. (b) Use part (a), the...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Use the third and fifth entries in Table 14.1.1 to...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The heat equation for the temperature u ( x , t ) .

Solution Summary: The author explains the solution of the heat equation for the temperature u(x,t).

Concept explainers

Videos

To solve: The heat equation for the temperature u(x,t).

Want to see the full answer?

Chapter 14 Solutions