- Write down the MATLAB commands for each of the following tasks.
(a) Evaluate
X
101
i=1
i
2
2π
sin
2π(i − 1)
100
.
(Hint: sum)
(b) Construct a 50 × 50 matrix, A, and fill in even numbers from 2 to
5000 column-wise, i.e. the first column is 2, 4, 6, 8, · · ·, the second
is 102, 104, 106, 108, · · ·, and so on. (Hint: reshape)
(c) Create a new matrix B containing the even rows and the odd columns
of A.
(d) Construct the following matrix
C =
1 −1 0 · · · · · · 0
−1 2 −1 0 · · · 0
0 −1 3 −1 0 · · ·
.
.
.
.
.
.
.
.
.
0 · · · 0 −1 119 −1
0 · · · · · · 0 −1 120
.
(Hint: diag)
(Note: You need to submit the m-file containing all the MATLAB commands for the above tasks.)
1 - The following differential equations describe the motion of a spaceship in
orbit about the earth (located at the origin) and the moon located at (1,0).
The derivatives are respect to time:
x
00 = 2y
0 + x −
µ0(x + µ)
r
3
1
−
µ(x − µ0)
r
3
2
− fx0
,
y
00 = −2x
0 + y −
µ0y
r
3
1
−
µy
r
3
2
− fy0
,
with
µ =
1
82.45
, µ0 = 1 − µ , r1 =
p
(x + µ)
2 + y
2 , r2 =
p
(x − µ0)
2 + y
2 .
(a) By plotting the orbit y(t) versus x(t) with initial conditions
x(0) = 1.2 , x0
(0) = 0 , y(0) = 0 , y0
(0) = −1.04935751 ,
show that when f = 0, the orbit is periodic with the period T =
6.19216933. You may use the MATLAB function ode45 to solve the
ODE numerically.
(b) With the same initial conditions as in (a), but with f = 1, solve the
differential equations from t = 0. Plot the orbit to show that the
spaceship eventually crashes.
(c) Repeat the computation of (b) with f = 0.1. Can you guess what is
happening?
(Note: You need to submit the m-files for producing the plots in (a),
(b) and (c) and the image files of the plots. Also, please provide the
description of what happens to the spaceship that you can see from the
plot of (c).)
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