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Math Help - Second Order Partial Differential Equation (Wave Eqn)

  1. #1
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    Exclamation Second Order Partial Differential Equation (Wave Eqn)

    Hi,
    i have this question:

    u_{tt} = c^2u_{xx}

    Find u(x,t) the deviateion from equilibrium for a stretched string fixed at its ends x=0 and x= pi. Initial conditions u(x,0)=alpha(sin(x) + 0.2sin(3x)), u_{t}(x,0)= 0.

    I know to use seperation of variables u(x, t) = F(x)G(t) giving me two ode's

    G"(t) - sG(t) = 0
    F"(x) - (s/ c^2)F(x) = 0 where s is a constant.

    This gives the solutions:
    F(x) = A1sin(nx)
    G(t) = A2cos(nct) + B2sin(nct)

    Now i'm stuck, how do I use the boundary and initial conditions to solve this? I know it has something to do with the fourier series?

    Please help,
    Katy
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by harkapobi View Post
    Hi,
    i have this question:

    u_{tt} = c^2u_{xx}

    Find u(x,t) the deviateion from equilibrium for a stretched string fixed at its ends x=0 and x= pi. Initial conditions u(x,0)=alpha(sin(x) + 0.2sin(3x)), u_{t}(x,0)= 0.

    I know to use seperation of variables u(x, t) = F(x)G(t) giving me two ode's

    G"(t) - sG(t) = 0
    F"(x) - (s/ c^2)F(x) = 0 where s is a constant.

    This gives the solutions:
    F(x) = A1sin(nx)
    G(t) = A2cos(nct) + B2sin(nct)

    Now i'm stuck, how do I use the boundary and initial conditions to solve this? I know it has something to do with the fourier series?

    Please help,
    Katy
    See here. Focus on the explanation to Problem A and make modifications to fit your problem.
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