Explanation The equations (1)-(4) are given as, ∂ 2 u ∂ t 2 = α 2 ∂ 2 u ∂ x 2 ... ( 1 ) The boundary conditions for this equation is, u ( 0 , t ) = u ( L , t ) = 0 ... ( 2 ) The initial conditions are, u ( x , 0 ) = f ( x ) ... ( 3 ) and ∂ u ∂ t ( x , 0 ) = g ( x ) ... ( 4 ) It is given that g ( x ) = { v 0 x a , 0 < x ≤ a , v 0 ( L − x ) ( L − a ) , a < x < L , and f ( x ) = 0 . The given equations are wave equation and it is known that the solution of a wave equation is given by, u ( x , t ) = ∑ n = 1 ∞ [ a n cos n π α L t + b n sin n π α L t ] sin n π x L ... ( 5 ) where, a n = 2 L ∫ 0 L f ( x ) sin n π x L d x b n = 2 n π α ∫ 0 L g ( x ) sin n π x L d x Since f ( x ) = 0 , a n = 0 . Now solve for b n , Further solve, b n = 2 n π α [ v 0 L 2 a n 2 π 2 sin n π a L − v 0 L n π cos n π a L + v 0 L − a [ ( L ( L − a ) n π cos n π a L ) + L 2 n 2 π 2 sin n π a L ] ] b n = 2 n π

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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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…

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Chapter 10.6, Problem 6E

To determine

The formal solution to the given initial value problem f(x)=0 and g(x)={v0xa,0<x≤a,v0(L−x)(L−a),a<x<L,.

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Chapter 10.6, Problem 5E

Chapter 10.6, Problem 7E

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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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The formal solution to the given initial value problem f ( x ) = 0 and g ( x ) = { v 0 x a , 0 < x ≤ a , v 0 ( L − x ) ( L − a ) , a < x < L , .

Solution Summary: The author explains that the formal solution to the given initial value problem is f(x)=0, and

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The formal solution to the given initial value problem f(x)=0 and g(x)={v0xa,0<x≤a,v0(L−x)(L−a),a<x<L,.

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Chapter 10 Solutions

The formal solution to the given initial value problem f ( x ) = 0 and g ( x ) = { v 0 x a , 0 &lt; x ≤ a , v 0 ( L − x ) ( L − a ) , a &lt; x &lt; L , .

Solution Summary: The author explains that the formal solution to the given initial value problem is f(x)=0, and

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The formal solution to the given initial value problem f(x)=0 and g(x)={v0xa,0<x≤a,v0(L−x)(L−a),a<x<L,.

Want to see the full answer?

Chapter 10 Solutions

The formal solution to the given initial value problem f ( x ) = 0 and g ( x ) = { v 0 x a , 0 < x ≤ a , v 0 ( L − x ) ( L − a ) , a < x < L , .