Chapter 4.P2, Problem 3P

Subject to the initial conditions $x\left(0\right)=0,x\text{'}\left(0\right)=1$, draw the graph of the numerical approximation of the solution of $\text{Eq}\text{.}\left(\text{1}\right)$ for $0\le t\le 30$. On the same set of coordinates axes, draw the graph of the solution of the linearized equations using the same initial conditions. Do this for each of the following sets of parameter values:

i.     $m=1,L=1,k=1,\text{and}h=1$

ii.   $m=1,L=1,k=1,\text{and}h=0.5$

iii.  $m=1,L=1,k=1,\text{and}h=0.1$

Compare the accuracy of the period length and amplitude exhibited by the solution of the linearized equation to the period length and amplitude of the solution of the nonlinear equation as the value of $h$ decreases.

$m{x}^{″}+f\left(x\right)=0,$ $\left(1\right)$