It turns out that the function
f
(
x
) =
x
m
is a solution to the
differential equation precisely when
0
2
x
(
mx
m
1
)
x
2
(
m
(
m
1)
x
m
2
(
m
(
m
3))
x
m
for all x. This is equivalent to
m
= 0 or
m
= 3.
______________________________________________________________________
3. (5 pts.)
It is known that every solution to the differential equation
y
″

y
= 0 is of the form
.
Which of these functions satisfies the initial conditions
y
(0) = 2 and
y
′
(0) = 8 ??
The initial conditions lead to the system of equations
which is equivalent to
The solution to the IVP is given by
.
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______________________________________________________________________
______________________________________________________________________
4. (10 points)
The following differential equation may be solved by either performing a substitution to reduce it
to a separable equation or by performing a different substitution to reduce it to a homogeneous equation. Display the
substitution to use and perform the reduction,
but do not attempt to solve the separable or homogeneous equation you
obtain
.
(5
x
2
y
1)
dx
(2
x
y
1)
dy
0
is equivalent to
The substitution is
x
=
X
+ 1 and
y
=
Y
 3. The reduction results in the
homogeneous DE
(5
X
2
Y
)
dX
(2
X
Y
)
dY
0
_________________________________________________________________
Bonkers 10 Point Bonus:
(a) The Fundamental Theorem of Calculus provides a neat formal solution involving
a definite integral with respect to the variable
t
to the following dinky IVP:
y
(
x
)
e
x
2
and y
(0)
1.
What is that solution? (b) Unfortunately the function
g
(
x
)
e
x
2
cannot be integrated in elementary terms. Use the answer to (a), the Maclaurin series for e
x
, and termbyterm integration,
to obtain a power series solution to the IVP. Write your answer using sigma notation. [Say where your work is! You don’t have
room here!]
(a)
y
(
x
)
1
⌡
⌠
x
0
e
t
2
dt for all x
.
(b)
y
(
x
)
1
⌡
⌠
x
0
e
t
2
dt
1
⌡
⌠
x
0
∞
k
0
(
t
2
)
k
k
!
dt
1
∞
k
0
⌡
⌠
x
0
(
t
2
)
k
k
!
dt
1
∞
k
0
⌡
⌠
x
0
t
2
k
k
!
dt
1
∞
k
0
x
2
k
1
(2
k
1)
k
!
for all x
.
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 Fall '08
 STAFF
 Derivative

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