Consider the pendulum system

l

$$ $$ g sin $=0(1)$

$1.$ Write the system in $$x $=$ f$($x$)$ form.

$2.$ Linearize the system around $(0,0)$ to get $$ $=$ A$.$ Explain in words the behavior of the system response

near $(0,0).$

$3.$ The Euler approximation of $$x $=$ f$($x$)$ is xk$+1=$ xk $+$tf$($xk$).$ Let $$t $=0\mathrm{.}1$ sec and and x$(0)=$

rand$(2,1).$ Compute the solution for k $=1,2,...,100.$

$4.$ The exact solution of the linearized system $$ $=$ A is k$+1=$ e

A$$t

k$.$ Let $$t $=0\mathrm{.}1$ sec and use the

same initial condition generated in the previous part. Compute the solution for k $=1,2,...,100.$

$5.$ The Matlab routine ode$45$ implements a high$-$accuracy Runge$-$Kutta formula to compute the solution

of differential equations. Let $$t $=0\mathrm{.}1$ sec and use the same initial condition generated in the previous

part. Compute the solution for k $=1,2,...,100$ using the ode$45$ routine in Matlab.

$6.$ Comment on the accuracy of the solution computed by each method. Can you do it in Matlab?