4. (15 pts.)
Suppose
is a solution of the homogeneous second order linear equation
Very neatly obtain the recurrence formula(s) needed to determine the
coefficients of
y
(
x
).
DO NOT WASTE TIME ATTEMPTING TO GET THE NUMERICAL
VALUES OF THE COEFFICIENTS.
First,
for all
x
near zero.
From this you can deduce that we have
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5. (10 pts.) (a) Suppose that
f
(
t
) is defined for
t
≥
0.
What is the
definition of the Laplace transform of
f
in terms of a definite integral??
for all
s
for which the integral converges.
(b) Using only the definition, not the table, compute the Laplace transform
of
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6.
(10 pts.)
Compute
when
(a)
(b)
’Tis just the usual magic of multiplication by ’1’ in the correct form or
the addition of ’0’ suitably transmogrified with linearity tossed into the
mix.
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7.
(5 pts.)
Locate and classify the singular points of the following
second order homogeneous O.D.E.
Use complete sentences to describe the
type of points and where they occur.
An equivalent equation in standard form is
From this, we can see easily that
x
0
= 0 is an irregular singular point of
the equation, and
x
0
= 4 is a regular singular point.
All other real
numbers are ordinary points of the equation.
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 Fall '08
 STAFF
 Derivative, #, 5 pts, 10 pts, 4 W

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