I'm trying to investigate nonlinear system numerical methods. For the nonlinear DE x' = 2t(1+x^2). Use the value tan(1) = 1.557407724654....
a) how to find the explicit solution $x(t)$ satisfying $x(0) = 1$?
b) how to use Euler's method to approx the value of $x(1) = e$ using $\Delta t = 0.1$. I.e., recursively determine $t_k$ and $x_k$ for $k = 1,...10$ with $\Delta t = 0$, starting with $t_0 = 0$ and $x_0 = 1$.
c) Repeat using $\Delta t = 0.05$
d) Again using Euler's method but reduce step size by a factor of 5, so that delta $t = 0.01$ to approx x(1)
e) Repeat parts b, c, and d with Improved euler's method using the same step sizes
f) Repeat using Runge-Kutta
g) Calculate the error in each case, since we now have 9 different approx for the value of $x(1) = e$, three for each method.
h) Calculate how the error changes as we change the step size from 0.01 to 0.05 and then from 0.05 to 0.01
Can someone also write in the format that implicit EUler and RK 4 takes in this example?