B U Department of Mathematics
Math 202 Differential Equations
Summer 2002 First Midterm
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1. Show that the differential equation
2
xy
3
dx
+ (3
x
2
y
2
+
x
2
y
3
+ 1)
dy
= 0
,
is not exact.
Find an integrating factor and a oneparameter family of solutions for this
equation.
Solution:
M
= 2
xy
3
,
N
= 3
x
2
y
2
+
x
2
y
3
+ 1
M
y
= 6
xy
2
,
N
x
= 6
xy
2
+ 2
xy
3
⇒
M
y
=
N
x
;
DE is not exact.
N
x

M
y
= 2
xy
3
=
M
⇒
An integrating factor
μ
=
μ
(
y
) exists:
μ
μ
=
N
x

M
y
M
= 1
⇒
μ
(
y
) =
e
y
Let
ψ
=
ψ
(
x, y
) be such that
ψ
x
= 2
xy
3
e
y
, ψ
y
= (3
x
2
y
2
+
x
2
y
3
+ 1)
e
y
ψ
x
= 2
xy
3
e
y
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 Winter '11
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 Equations, Trigraph

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