2
.
(b)
An ODE for the orthogonal trajectories is now given by
,
dy
dx
x
2
2
y
x
or equivalently,
.
dy
dx
2
x
y
x
.
(c)
This little first order linear equation has
µ
=x
2
as an integrating
factor.
Using the standard recipe, a oneparameter family of solutions is
given by
.
x
2
y
x
4
4
K
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______________________________________________________________________
7.
(10 pts.) It turns out that the nonzero function f(
x
) = exp(
x)
is a
solution to the homogeneous linear O.D.E.
(*)
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.
(a)
If we have
,
y
ve
x
then
,
y
e
x
v
e
x
v
,
y
e
x
v
2
e
x
v
e
x
v
and
.
y
e
x
v
3
e
x
v
3
e
x
v
e
x
v
Substituting
y
into (*) and then replacing
v
′
using
w
implies that
w
must
be a solution to
(**)
w
3
w
3
w
0.
(b)
The auxiliary equation for (**) above is
m
2
3
m
3
0.
By using quadratic formula, it is easy to see that the roots are
m
3
2
±
3
2
i
The general solution to (**), then, is
w
c
1
cos(
3
2
x
)
e
3
2
x
c
2
sin(
3
2
x
)
e
3
2
x
.
(c)
The equation
v
′
=
w
is a constant coefficient linear ODE with
w
being a UC function. [
Look at w above, Folks!
]
Consequently, we can
completely solve this ODE without performing any actual integrations.
_________________________________________________________________
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.
[Hints are on Page 2 of 4.]
(a)
y
y
0.
(b)
(sin(
x
)
x
cos(
x
))
y
(
x
sin(
x
))
y
(sin(
x
))
y
0.
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 Fall '08
 STAFF
 Vector Space, exv

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