, of the differential equation
y
y
4sec(
x
).
______________________________________________________________________
4.
(10 pts.)
Set up the correct linear combination of undetermined coefficient functions you would use to find a
particular integral, y
p
, of the O.D.E.
y
4
y
5
y
x
2
e
2
x
sin(
x
)
e
2
x
.
[
Warning:
(a) If you skip a critical initial step, you will get no credit!! (b) Do not waste time attempting to find the
numerical values of the coefficients!!
]
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TEST2/MAP2302 Page 3 of 4
______________________________________________________________________
5.
(10 pts.) The factored auxiliary equation of a certain homogeneous
linear O.D.E. with real constant coefficients is as follows:
m(m  2
π
)
2
(m  (5i))
3
(m  (5i))
3
= 0
(a) (5 pts.) Write down the general solution to the differential equation.
[
WARNING: Be very careful. This will be graded Right or Wrong!!
]
(b) (5 pt.) What is the order of the differential equation?
______________________________________________________________________
6. (15 pts.)
(a) Obtain the differential equation satisfied by the
family of curves defined by the equation (*) below.
(b) Next, write down the differential equation that the
orthogonal trajectories to the family of curves defined by (*) satisfy.
(c) Finally, solve the differential equation of part (b) to
obtain the equation(s) defining the orthogonal trajectories. [These, after
all, are another family of curves.]
(*)
.
x
2
2
y
1
ce
2
y
TEST2/MAP2302 Page 4 of 4
______________________________________________________________________
7.
(10 pts.) It turns out that the nonzero function f(
x
) = exp(
x)
is a
solution to the homogeneous linear O.D.E.
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 Fall '08
 STAFF
 Vector Space, Partial differential equation

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