linearization of 2nd order differential equations

can anyone help with these problems?

In week 4, we showed how to set up the second order difierential equation which describes
the motion of the peadulum. Unfortunately, the di$erential quation is nonlinear, so at that
time, we linearized about 0 :0 which mealls that the solution will be good only for small
angles. Recatl that the differentia,l we discussed in week 4 is

#.*X+fsin(d):o
Fbr our project, we will let m : t, L : 1, c : 0.5, and recall g : 9.8. Therefore, our
difierential equation is

&g
m + r.u#* e.8sin(o)

: Q
1) (10 pts) Show the steps in using the change ofvariables

to convert the above nonlinear, seond order differential equation (the one with mrmbe.rs) to
the fust order system

dr
dt:g
y : *g.8 sin(r) – 0.5ydt

fi T 0

a: f

e) (5 pts) Lhearize the nonlinear system about the critical point (n,0) and determine stabil-
ity. This means to actually write down the linearized system about (z’,0) (so you will need
to find partial derivatives and evaluate at, r: zr and U: A), and theu use the eigenvalues of
the coefficient rnatrix to determine stability.

d) (5 pts) tinearize the aonlinear systemn about the critical point (22r,0) and determine
stability. This means to actually write dovrn the linearized systern about (2zr,0) (* yo*
will need to fiad partial derinatives and erialuate at s : 2r and g : 0), and then use the
eigenvalues of the coefficient matrix to determine stability.

4) (20 points)
a) (5 pts) \ferify that (0,0), (o,0), and (2r,0) are dl critical points of the first order system
(indeed, all (mr,0) will be critical points).

b) (5 pts) Linearize the uonlinear systenr about the critical point (0,0) and determine stabil-
ity” This means to actually write down tbe linearized system about (0,0) (so 1’ou will need
to find partial derirnatives and evaluate at s:0 and U:0), and then use the eigeavalues of
the coefficient matrix to determine stability.

3) (10 pts) I used NumSysDE.:uncd to numerically solve the difierential equation with the
given initial conditions and then created the graphs of r vs t in each case (keep in mind that
r:0).

“(0) ==
0, y(0) – $ r(0) : 0, gr(0) =- $

Using the graphs to think about the actual motion of the pendulum, e,rrplain why the graphs
appear as they do. In particular, you should be able to explain whf Egr(t)

: 2n in the
smond Saph.

2) (10 pts) I used NumSysDE.xmcd to mrmerically solve the differential equation with the
initial conditions 0(0) : 0 and d (t1 :5 (i.e. z(O) : 0 aud y(0) : b) and then created rhe
graph of 0 vs , (k”up io mind that s : CI).

Use this Saph to discuss the actual motion of the pendulum in the time interval A 3t < 4. That is, where irs the pendulum at time t : 0? Then as time increases, is the peuduhrm moving cloclrwise or countercloclrwise and for approximately how long? ETC.