Symbolically solving Wikipedia Runge-Kutta example?


by honglin
Tags: rungekutta, solving, symbolically, wikipedia
honglin
honglin is offline
#1
May7-12, 05:51 PM
P: 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?


Notes:
[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!
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