Obtain the differential equation satisfied by
the family of curves defined by the equation (*) below.
(b)
(3 pts.)
Next, write down the differential equation
that the orthogonal trajectories to the family of curves defined by (*)
satisfy.
(c)
(3 pts.)
Finally, solve the differential equation of
part (b) to obtain the equation(s) defining the orthogonal trajectories.
(*)
y
e
cx
(a)
(b)
(c)
The differential equation in part (b) has no constant solutions, when
y
is considered a function of
x
, and separating variables yields
.
Integrating, and clearing the common denominator provides us with
,
an equation for the orthogonal trajectories.
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______________________________________________________________________
6.
(10 pts.) The equation
has a regular singular point at
x
0
= 0.
Substitution of
into the ODE and a half a page of algebra yields
Using this information, (a)
write the form of the two linearly independent
solutions to the ODE given by Theorem 6.3 without obtaining the numerical
values of the coefficients of the series involved, and then (b) do obtain
the the numerical values of the coefficients
c
1
, ... ,
c
5
to the first
solution,
y
1
(
x
), given by Theorem 6.3 when
c
0
= 1. [
Keep the parts separate.
]
(a)
Plainly, the indicial equation is
r
2
 1 = 0, with two roots
r
1
= 1
and
r
2
= 1.
Consequently the two linearly independent solutions provided
by Theorem 6.3 look like the following:
and
.
(b)
When
r
=
r
1
= 1, (1+
r
)
2
 1
≠
0.
Thus, we must have
c
1
= 0, and
Using this recurrence and the given information, it follows that
______________________________________________________________________
7.
(10 pts.) Solve the following second order initialvalue problem using
only the Laplace transform machine.
By taking the Laplace Transform of both sides of the differential
equation, and using the initial conditions, we have
By taking inverse transforms now, we quickly obtain
MAP2302/FinalExam Page 6 of 6
______________________________________________________________________
8.
(10 pts.) (a)
Given
f
(
x
) =
x
is a nonzero solution to
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 Fall '08
 STAFF
 Ode

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