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!
Phys.Org News Partner Science news on Phys.org
Better thermal-imaging lens from waste sulfur
Hackathon team's GoogolPlex gives Siri extra powers
Bright points in Sun's atmosphere mark patterns deep in its interior

Register to reply

Related Discussions
Solving Second Order Differential Equations using Runge Kutta Calculus & Beyond Homework 7
Solving the Kinematic Equations using Runge-Kutta Engineering, Comp Sci, & Technology Homework 2
Runge Kutta for solving 2nd order ODE Introductory Physics Homework 0
Runge-Kutta or Runge-Kutter? Differential Equations 5
Runge kutta method for solving PDE Differential Equations 0