Exercises 1.3
11.
For this equation, the isoclines are given by 2
x
=
c
. These are vertical lines
x
=
c/
2. Each
element of the direction field associated with a point on
x
=
c/
2 has slope
c
. (See Figure B.7
in the answers of the text.)
13.
For the equation
∂y/∂x
=
−
x/y
, the isoclines are the curves
−
x/y
=
c
. These are lines that
pass through the origin and have equations of the form
y
=
mx
, where
m
=
−
1
/c
,
c
= 0. If
we let
c
= 0 in
−
x/y
=
c
, we see that the
y
axis (
x
= 0) is also an isocline. Each element
of the direction field associated with a point on an isocline has slope
c
and is, therefore,
perpendicular to that isocline. Since circles have the property that at any point on the circle
the tangent at that point is perpendicular to a line from that point to the center of the circle,
we see that the solution curves will be circles with their centers at the origin. But since we
cannot have
y
= 0 (since
−
x/y
would then have a zero in the denominator) the solutions will
not be defined on the
x
axis. (Note however that a related form of this differential equation is
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Slope, Conic section

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