Explanation Proof: Consider the given equations, ∂ 2 u ∂ t 2 = α 2 ∂ 2 u ∂ x 2 − ∞ < x < ∞ , t > 0 …… (1) u ( x , 0 ) = f ( x ) − ∞ < x < ∞ , …… (2) ∂ u ∂ t ( x , 0 ) = g ( x ) − ∞ < x < ∞ , …… (3) The solution of equation (1) is given by formula u ( x , t ) = A ( x + α t ) + B ( x − α t ) …… (4) The requirement is to only choose the functions A and B so that the initial condition that is equation (3) and equation (2) are satisfied. Therefore, u ( x , 0 ) = A ( x ) + B ( x ) = f ( x ) …… (5) ∂ u ∂ t ( x , 0 ) = α A ' ( x ) − α B ' ( x ) = g ( x ) …… (6) Integrate equation (6) take lower limit as x 0 and upper limit as x and divide the equation by α . A ( x ) − B ( x ) = 1 α ∫ x 0 x g ( s ) d s + C …… (7) C is an arbitrary constant. Now, solve equation (5) and (6)

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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 10.6, Problem 12E

To determine

To proof:

The D’Alembert’s solution to the initial value if f(x) has continuous second derivative and g(x) has continuous first derivative by substitute it directly into the equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.6, Problem 11E

Chapter 10.6, Problem 13E

Students have asked these similar questions

Find the indefinite integral. (Use C for the constant of integration.) 9x arcsin(x) dx

Find the indefinite integral using the substitution x = 5 sin(e). (Use C for the constant of integration.) 1 dx (25-x²)3/2

Find the indefinite integral using the substitution x = 7 sec(0). (Use C for the constant of integration.) √ ×³ √x² - 49 dx

Chapter 10 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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.

To proof: The D’Alembert’s solution to the initial value if f ( x ) has continuous second derivative and g ( x ) has continuous first derivative by substitute it directly into the equation.

Solution Summary: The author explains D'Alembert's solution to the initial value by substituting it directly into the equation.

Concept explainers

Videos

To proof:

The D’Alembert’s solution to the initial value if f(x) has continuous second derivative and g(x) has continuous first derivative by substitute it directly into the equation.

Want to see the full answer?

Chapter 10 Solutions

To proof: The D’Alembert’s solution to the initial value if f ( x ) has continuous second derivative and g ( x ) has continuous first derivative by substitute it directly into the equation.

Solution Summary: The author explains D'Alembert's solution to the initial value by substituting it directly into the equation.

Concept explainers

Videos

To proof: The D’Alembert’s solution to the initial value if f(x) has continuous second derivative and g(x) has continuous first derivative by substitute it directly into the equation.

Want to see the full answer?

Chapter 10 Solutions

To proof:

The D’Alembert’s solution to the initial value if f(x) has continuous second derivative and g(x) has continuous first derivative by substitute it directly into the equation.