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    Basically I'm doing a research project-type thing and it's led me to a coupled differential equation looking like

    x_{i+1}''(t)=\nu (x_i'(t)-x_{i+1}'(t))

    I have initial conditions x_i(0)=A_i and x_i'(0)=0 and a known function for x_1(t).

    I could recursively go through the ODEs and calculate (or get MATLAB to do it) the solutions, but this is rather painstaking and I can't remember/don't know how I would generate general solutions for this particular ODE.

    Any help is much appreciated.
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    There is always recursion, but you can integrate to express the solutions recursively.

    integrating once gives
    x'_{i+1} + \nu x_{i+1} = \nu (x_i - A_i + A_{i+1})

    Integrating factor, and integrate again:
    x_{i+1}(t) = \int_0^t \nu e^{\nu(\tau-t)}x_i(\tau)d\tau + A_{i+1} - A_i(1-e^{-\nu t}).
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    (Original post by liewchuan)
    There is always recursion, but you can integrate to express the solutions recursively.

    integrating once gives
    x'_{i+1} + \nu x_{i+1} = \nu (x_i - A_i + A_{i+1})

    Integrating factor, and integrate again:
    x_{i+1}(t) = \int_0^t \nu e^{\nu(\tau-t)}x_i(\tau)d\tau + A_{i+1} - A_i(1-e^{-\nu t}).
    I've done the first couple of iterations recursively by hand, though many thanks for the help.
 
 
 
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