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

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 10.6, Problem 12E

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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.

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Chapter 10.6, Problem 11E

Chapter 10.6, Problem 13E

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Chapter 10 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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...

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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.

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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.