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Homework 1
AMATH 383, Autumn 2009
Due: Friday, October 16
1. (each 1 point) For the following equations determine the order and if it is nonlinear or linear:
(a)
y
0

5
y
2
+
y
= 0
,
(b)
y
00
y
+
y
0
+ 2
y
= 0 (c)
6
y

y
0
y
00
= 1
(d)
y
0
(
x
) = (2 +
x
)
y
(
x
) (e)
d
4
x
dt
4
+ 2
t
dx
dt
= exp(
x
)
(f)
d
dx
±
y
00
(
x
) +
xy
(
x
)
y
0
(
x
)
¶
= 1
Solution:
(a)
nonlinear, 1st order
(b)
nonlinear, 2nd order
(c)
linear, 2nd order
(d)
linear, 1st order(nonconstant coeﬃcients)
(e)
nonlinear, 4th order
(f)
nonlinear, 3rd order
2. (4+4+7 points) Give the general solutions to the following equations:
(a)
y
00
=

3
y
0
,
(b)
6
y

y
0
y
00
= 1
(c)
y
0
= 6
y

4
y
2

2
Solution:
(a)
y
(
x
) =
C
1
+
C
2
e

3
x
C
1
, C
2
= const (Ansatz
e
λx
)
(b)
y
(
x
) =
C
1
e

3
x
+
C
2
e
2
x
C
1
, C
2
= const (Ansatz
e
λx
)
(c)
separation of variables:
Z
d
x
=
Z
1
6
y

4
y
2

2
d
y
=
Z ±
1
2
y

1

1
/
2
y

1
¶
d
y
⇒
ln(2
y

1)

1
2
ln(
y

1) =
x
+
C
⇒
y
(
x
) =
˜
Ce
2
x

1
˜
Ce
2
x

2
3. (2+2 points) The following gives equivalent solutions to ODEs which written with diﬀerent
integration constants. Give a relation between the constants.
1
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View Full Document Homework, AMATH 383, Autumn 2009
(a) For
y
0
=
y
(1

y
) equivalent solutions are
y
(
x
) =
C
exp(
x
)
1 +
C
exp(
x
)
=
1
1 +
ˆ
C
exp(

x
)
=
exp(
x
+
˜
C
)
exp(
x
+
˜
C
)

1
(b) For
y
00
+
ω
2
y
= 0 equivalent solutions are
y
(
x
) =
C
1
exp(
ω
i
x
) +
C
2
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This note was uploaded on 01/10/2010 for the course MATH 124 taught by Professor Walker during the Spring '08 term at University of Washington.
 Spring '08
 WAlker
 Calculus, Equations

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