Group Project
MAP 2302
In this project, we’ll reexamine massspring systems, but with variable coefficients. But first we
will help show that Einstein’s relativity formulas generalize kinetic energy in the classical sense.
That is, if the speed of an object is small compared to the speed of light, then the relativistic result
is extremely close to
1
2
mv
2
.
1.
We first find a particular power series that will help with our calculation. . . the binomial series.
(a) Use the method of integrating factors from Unit 1 to find
(1 +
x
)
α
to be a solution to the
IVP below. Is it unique?
(1 +
x
)
y
0

α y
= 0 ;
y
(0) = 1
(Here
, α
is any fixed real number
.
)
(b) Now we use the method of power series find a solution of the form
∞
X
n
=0
c
n
x
n
. Find a
formula for
c
n
+1
in terms of
c
n
. Show that this lead to the power series:
1 +
∞
X
n
=0
α
(
α

1)
. . .
(
α

(
n

1))
n
!
x
n
= 1 +
α x
+
α
(
α

1)
2!
x
2
+
α
(
α

1)(
α

2)
3!
x
3
. . .
(c) What is the radius of convergence of this series by ratio test? How does this compare
with the minimum radius of convergence guaranteed by the differential equation? Is
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 Spring '08
 TUNCER
 Derivative, Power Series, Taylor Series, Mathematical Series

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