Symbolically solving Wikipedia Runge-Kutta example?

[honglin]

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!

 

[jvicens]

Thank you for bringing this to my attention. I am also a scientist and have experience with solving differential equations. After reviewing the DE provided in the example and attempting to solve it symbolically, I have come to the conclusion that this equation does not have a simple symbolic solution.

The reason for this is because the equation is a non-linear DE, meaning that the dependent variable (y) is not directly proportional to the independent variable (t). In these cases, it is often difficult to find a closed-form solution and numerical methods, such as the Runge-Kutta method, are used to approximate the solution.

In addition, the fact that the equation includes the trigonometric function tan(y) makes it even more complicated to solve symbolically. It is possible to use special functions, such as the Lambert W function, to express the solution, but it would still be a complex expression.

I hope this helps to clarify why the symbolic integration led to more complicated equations. Sometimes, numerical methods are the best approach for solving non-linear DEs. Thank you for bringing this interesting problem to my attention.