(a)
It is trivial
to obtain a 4th order homogeneous linear constant coefficient ordinary
differential equation with
f
and
g
as solutions.
Do so.
(b)
It’s only slightly messier to obtain a 2nd order homogeneous linear
ordinary differential equation with {
f
,
g
} as a fundamental set of
solutions.
Do so.
Hints:
(a)
Here one should expect that a fundamental set of solutions
should be { 1,
x
, sin(
x
), cos(
x
) }, arising from the auxiliary equation
m
2
(
m
2
+ 1) = 0.
The constant coefficient equation has a family
resemblance.
(b)
Substitute f and g into the ODE
y
p
(
x
)
y
q
(
x
)
y
0
to obtain a linear system in
p
(
x
) and
q
(
x
).
Find
p
and
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______________________________________________________________________
5.
(10 pts.) The factored auxiliary equation of a certain homogeneous
linear O.D.E. with real constant coefficients is as follows:
m
(
m2
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?
y
c
1
c
2
e
2
π
x
c
3
xe
2
π
x
c
4
sin(5
x
)
c
5
cos(5
x
)
c
6
x
sin(5
x
)
c
7
x
cos(5
x
)
c
8
x
2
sin(5
x
)
c
9
x
2
cos(5
x
)
The order of the differential equation is 9.
______________________________________________________________________
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
(a)
Differentiating (*) with respect to x and then replacing
c
yields
2
x
2
dy
dx
2
ce
2
y
dy
dx
dy
dx
2
2(
x
2
2
y
1)
e
2
y
e
2
y
dy
dx
4
y
2
x
2
.
Thus, a differential equation for the family of curves is given by
dy
dx
x
2
y
x
2
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 Fall '08
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 Vector Space, exv

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