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MAE 290B  Homework # 1
Numerical Methods in Science and Engineering
Prof. Alison Marsden
Due date: Thurs Jan 20, 2011
Problem 1  Euler Method.
A physical phenomena is governed by the linear diﬀerential
equation
dv
dt
=

0
.
2
v

2 cos(2
t
)
v
2
subject to the initial value
v
(0) = 1.
(a) Solve this equation analytically.
(b) Write a program to solve the equation for 0
< t
≤
7 using the explicit Euler scheme with
the following time steps:
h
= 0
.
2
,
0
.
05
,
0
.
025
,
0
.
006. Plot the four numerical solutions
along with the exact solution on one graph with axes 0
< x <
7 and 0
< y <
1
.
4.
Discuss your results.
(c) In practical problems, the exact solution is not always available. To obtain an accurate
solution, we keep reducing the time step (usually by a factor of 2) until two consecutive
numerical solutions are nearly the same. Assuming that you do not know the exact
solution for the present equation, do you think that the solution corresponding to
h
= 0
.
006 is accurate (to plotting accuracy)? Justify your answer. In case you ﬁnd it
not accurate enough, obtain a better one.
Problem 2  Apollo Orbit.
The following diﬀerential equations describe the motion of
a body in orbit about two much heavier bodies. An example would be an Apollo capsule in
an earthmoon orbit. The three bodies determine a plane in space and a twodimensional
Cartesian coordinate in this plane. The origin is at the center of mass of the two heavy
bodies, the
x
axis is the line through these two bodies, and the distance between them is
taken as the unit. Thus, if
μ
is the ratio of the mass of the moon to that of the earth, then
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This note was uploaded on 03/13/2011 for the course MAE 290B taught by Professor Marsden during the Winter '11 term at UCSD.
 Winter '11
 MARSDEN

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