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Answer Guide 1
Math 427K: Unique Number 55160
Wednesday, August 31, 2011
1. Suppose that the area bounded by a smooth curve
y
(
x
), the
x
axis, the
y
axis, and the
vertical line through
x
equals twice the curve’s arc length between 0 and
x
.F
i
n
dt
h
e
diferential equation satis±ed by the curve.
Solution:
Writing
°
x
0
y
(ˆ
x
)dˆ
x =2
°
x
0
±
1+(
y
°
(ˆ
x
))
2
dˆ
x
and applying the ²undamental theorem o² calculus, one obtains
y
(
x
)=2
±
1+(
y
°
(
x
))
2
.
Simpli²ying, we get
(
y
°
(
x
))
2
=
y
(
x
)
2
4
−
1
.
(
°
)
2. A curve in the
x, y
plane is de±ned by the condition that the sum o² the
x
 and
y
intercepts
o² its tangents always equals
m
. Find a diferential equation ²or the curve.
Solution:
Note that a correct answer here may be written in more than one ²orm.
The tangent line so a smooth curve
y
(
x
) at an arbitrary point (
a, b
) has the ²orm
y
−
b
=
y
°
(
a
)(
x
−
a
)
.
The
y
intercept, ²ound by setting
x
= 0, is
y
int
(
a, b
)=
b
−
ay
°
(
a
)
.
The
x
intercept, ²ound by setting
y
= 0, is
x
int
(
a, b
)=
a
−
b
y
°
(
a
)
.
(You may assume that
y
°
(
a
) is never zero, since otherwise the
x
intercept does not exist.)
Note that we have ²ound
y
int
and
x
int
as ²unctions o² the point (
a, b
). Since (
a, b
) was
arbitrary, we can substitute
x
=
a
,
y
=
b
, and
dy
dx
=
y
°
(
x
) to get a nicer ²ormula
m
=
x
²
1
−
dy
dx
³
+
y
²
1
−
dx
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This note was uploaded on 10/24/2011 for the course MATH 427K taught by Professor Delallave during the Spring '11 term at University of Texas at Austin.
 Spring '11
 DELALLAVE
 Differential Equations, Equations, Arc Length

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