(*)
y
y
0.
(a) Reduction of order with this solution involves making the substitution
y
ve
x
into equation (*) and then letting
w
=
v
′
. Do this substitution and obtain
the constant coefficient equation that
w
must satisfy. (b) Obtain a the
general solution to the ODE that
w
satisfies and then stop.
(c) Explain very briefly why
v
can be obtained from
w
without actually
integrating.
Do not attempt to actually find v.
_________________________________________________________________
Silly 10 Point Bonus:
Let
f
(
x
) =
x
and
g
(
x
) = sin(
x
). (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. [Say where your work is, for it won’t fit here.]
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 Fall '08
 STAFF
 Vector Space, Partial differential equation

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