Homework $10$: ODE Algorithms

$($due by email before the lecture on Mon, April $22)$

Write your own rk$2$ method. Design your method for a general ODE; this means making the derivative function $f(t,x)$ a separate method.

Use your $rk22$ solver in a program that solves the equation of motion $($a second$-$order ODE$)$ for a charged particle moving in a variable external electric field within an anharmonic trapping potential given by:

$m\frac{d^{2}x}{d{t}^2}=F_{\text{e}\text{x}\text{t}}(x,t)-k{x}^{p-1}$

where $p=4$ and $F_{\text{e}\text{x}\text{t}}=-qE{x}^2$; assume $q=m=k=1$ and $E=5$ for simplicity and assume that the motion starts at $x(0)=0$ with the particle velocity $x^{\text{'}}(0)=1.$

$3.$ Plot the position of the particle $x(t)$ for the first $10$ seconds of motion to $10).$ Find the turning points of the motion $($ min and max of the position, $x(t)).$

$4.$ Solve the same equation using the rk$4$ and rk$45$ solvers $($use the code in the course textbook and$/$or implementation in SciPy$).$ Plot the three solutions $($rk$2,$ rk$4,$ and rk$45)$ in one figure and the relative error of rk$2$ vs rk$4($or rk$45)$ in another figure.

$5.$ Time the three methods and report their execution times.