Homework 1
AMATH 383, Autumn 2009
Due: Friday, October 16, after class
1. (each 1 point) For the following equations determine the order and if it is nonlinear or linear:
(a)
y
′
−
5
y
2
+
y
= 0
,
(b)
y
′′
y
+
y
′
+ 2
y
= 0
(c)
6
y
−
y
′
y
′′
= 1
(d)
y
′
(
x
) = (2 +
x
)
y
(
x
)
(e)
d
4
x
dt
4
+ 2
t
dx
dt
= exp (
x
)
(f)
d
dx
p
y
′′
(
x
) +
xy
(
x
)
y
′
(
x
)
P
= 1
2. (4+4+7 points) Give the general solutions to the following equations:
(a)
y
′′
=
−
3
y
′
,
(b)
6
y
−
y
′
y
′′
= 1
(c)
y
′
= 6
y
−
4
y
2
−
2
3. (2+2 points) The following gives equivalent solutions to ODEs which written with di±erent
integration constants. Give a relation between the constants.
(a) For
y
′
=
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
′′
+
ω
2
y
= 0
equivalent solutions are
y
(
x
) =
C
1
exp (
ω
i
x
) +
C
2
exp (
−
ω
i
x
) =
ˆ
C
1
cos (
ω x
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 Spring '08
 WAlker
 Calculus, Equations, following equations, Homogeneity, equivalent solutions

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