Fourth Order Runge Kutta method Detailed Step by Step Soln.

Question
Use Runge-Kutta method of order four to approximate the solution to the given initial value problem and compare the results to the actual values.

y'=e^(t-y) , 0 <=t <=1 , y(0)=1 with h = 0.5(Interval)

Actual solution is y(t)= In((e^t+e-1).

For full description of the problem, please see the attached question file.

Solution:
For the first interval of h = 0.5 we get as below (for step by step explanation please see the attached solution file)
k1=0.183940 , k2=0.215430 , k3= 0.212065 , k4 = ...

Solution Summary

This solution is comprised of detailed explanation of using Runge Kutta method of order four to solve Ordinary Differential Equations(Initial Value Problems). Formulas included in standard notation and the solution is explained in easy to understand format. The attached solution file contains 5 pages in which each minute detail regarding the solution is given.
Students would be able to solve other problems on this topic easily with the help of this solution.
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m*dv/dt = mg - kv^, k>0.
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Need to solve and compare the results of the linear vs non linear pendulum problem. Compare the solutions for the approximation (linear) and numerical(non-linear) using the numerical methodrungekutta of 4th DEGREE OR HIGHER(preferrably 4th-6th order). Please include the following details:
1. Detailed explanation of methods

Write a general purpose RK routine; here described in a matlab
context (but feel free to re-interpret into and program in any language environment)
[t_out,y_out,e_out] = rk ( ode_RHS, y_0, t_range, c, A, b1, b2 )
input parameters:
ode_RHS= function handle to right side f of the ODE y'=f(t,y)
y_0= initial value for ODE

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dy/dx = 3xy²; y(0) = 1; 0 ≤ x ≤ 1

Question:
How does this compare with the theoretical rate of convergence of 0(H2). Explain your result as best as you can?
Data:
We used the Runge-Kuttamethod to solve
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See attached files for full problem description.

See the attached file.
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(see attached file for equations).
Even though these equations are simple and deterministic, long-term behavior of solutions for some particular values of parameters (e.g. omega = 10, ro = 28, beta = 8/3) could be highly unpred

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See attached for equations.
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