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

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Chapter 10.6, Problem 6E

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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 10.6, Problem 5E

Chapter 10.6, Problem 7E

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Fundamentals of Differential Equations and Boundary Value Problems

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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 , .