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Symbolically solving Wikipedia Runge-Kutta example?

  1. May 7, 2012 #1
    The Wikipedia page for "Runge-Kutta methods"[1] gives the following example:

    y' = tan y + 1
    y(1) = 1
    t in [1, 1.1]

    Using a step-size of h = .025, this solution is found:

    y(1.1) = 1.335079087

    I decided to check this solution by solving symbolically. But my attempts to symbolically integrate only lead to more complicated equations.[2] So I'm wondering if this simple-looking DE actually has a symbolic solution?

    [1] wikipedia (dot) org/wiki/Runge%E2%80%93Kutta_methods
    [2] For example,

    y'(t) = tan(y(t)) + 1

    y'(t)/(tan(y(t)) + 1) = 1

    Let u = y(t), du = y'(t) dt

    ∫(du/(tan(u) - 1)) = ∫dt

    I used the SAGE computer algebra system to evaluate the LHS to,

    -1/2*u + 1/2*log(tan(u) - 1) - 1/4*log(tan(u)^2 + 1)

    Not much help!
  2. jcsd
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