Differential Equations Quiz
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SYSTEMS OF DIFFERENTIAL EQUATIONS
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Consider the system
d
x
d
t
=
−
y
+
x
(
x
2
+
y
2
)
\frac{dx}{dt} = -y + x(x^2 + y^2)
d
t
d
x
=
−
y
+
x
(
x
2
+
y
2
)
,
d
y
d
t
=
x
+
y
(
x
2
+
y
2
)
\frac{dy}{dt} = x + y(x^2 + y^2)
d
t
d
y
=
x
+
y
(
x
2
+
y
2
)
. In polar coordinates
(
r
,
θ
)
(r, \theta)
(
r
,
θ
)
, what is the correct description of the trajectory behavior?
A
Trajectories spiral inward toward the origin since
d
r
d
t
=
−
r
3
<
0
\frac{dr}{dt} = -r^3 < 0
d
t
d
r
=
−
r
3
<
0
for all
r
>
0
r > 0
r
>
0
B
Trajectories spiral outward from the origin since
d
r
d
t
=
r
3
>
0
\frac{dr}{dt} = r^3 > 0
d
t
d
r
=
r
3
>
0
for all
r
>
0
r > 0
r
>
0
C
Trajectories form closed circular orbits since
d
r
d
t
=
0
\frac{dr}{dt} = 0
d
t
d
r
=
0
and
d
θ
d
t
=
1
\frac{d\theta}{dt} = 1
d
t
d
θ
=
1
D
Trajectories approach the unit circle since
d
r
d
t
=
r
(
1
−
r
2
)
\frac{dr}{dt} = r(1-r^2)
d
t
d
r
=
r
(
1
−
r
2
)
changes sign at
r
=
1
r = 1
r
=
1
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