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HW2_00Whint3

# HW2_00Whint3 - Analytical Solution of a Non-homogeneous PDE...

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Analytical Solution of a Non-homogeneous PDE for Vibration of an Elastic String Additional Note for Homework #2 February 13, 2000 We shall solve a vibration problem of an elastic string spanned on the interval ( ) 0,1 : ( ) ( ) ( ) ( ) 2 2 2 2 , , , 0,1 , u u f x t x t t x = × −∞ +∞ with the homogeneous initial condition ( ) ( ) ,0 ,0 0 u u x x t = = and the homogeneous boundary condition ( ) ( ) 0, 1, 0 u t u t = = . To do this, we first assume that the non-homogeneous term ( ) , f x t can be expressed by ( ) ( ) ( ) ( ) ( ) ( ) 1 0 1 , sin , 2 , sin k k k f x t f t k x f t f t k d π ξ πξ ξ = = = Then assuming the solution ( ) , u x t in the following form ( ) ( ) ( ) 1 , sin k k u x t u t k x π = = we shall find appropriate coefficient functions ( ) k u t . Substitution of this into the differential equation yields ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 2 2 2 1 1 sin sin , sin , , 0,1 , k k k k k k k u u u t k x t x t x d u t k u t k x dt f x t f t k x x t π π π π = = = = = + = = × −∞ +∞

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HW2_00Whint3 - Analytical Solution of a Non-homogeneous PDE...

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