Explanation Given: Length of string = L Interval of x -axis = [ 0 , L ] Function u ( x , t ) represents displacement at a point x along the string at time t . Procedure used: (1) Write the general equation. (2) Write the initial conditions of the problem. (3) Write the boundary conditions of the problem. Calculation: The general wave equation for string is given by, a 2 ∂ 2 u ∂ x 2 + F ( x , t , u , u t ) = ∂ 2 u ∂ t 2 where F is the net external force. Given that, an external force proportional to the horizontal distance from the left end acts on the string for t > 0. If the horizontal distance is denoted by x , then F ∝ x . In other words, F = A x , where A is a constant . Thus, the wave equation becomes, a 2 ∂ 2 u ∂ x 2 + A x = ∂ 2 u ∂ t 2 , 0 < x < L , t > 0 , A is a constant . The string is initially at rest on x -axis. Hence at t = 0 , the value of u ( x , t ) is 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

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ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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