MATH 215/255
Fall 2014
Assignment 9
§3.9, §8.1, §8.2
Solutions to selected exercises can be found at the end of the draft chapter of Lebl’s book
on nonlinear systems.
• 3.9.8: Consider the equation
→
−
x0 =
1/t −1
1 1/t
→
−
x +
t2
−t
.
Check that a complementary solution is
t sin t
t cos t
→
−
x c = c1
+ c2
.
−t cos t
t sin t
Use it to find a particular solution, by variation of parameters.
• 8.1.3 (modified): Find the critical points and the Jacobian matrix at any point for
the following systems.
0 2
0 0 x +y
x
−y
x
x2 − y 2
x
.
=
, c)
=
, b)
=
a)
y2 + x
y0
3x + yx2
y0
x2 + y 2 − 1
y0
• 8.1.4: For the following systems, find the linearization at (0, 0).
0 0 −y
x
x + 2y + x2 − y 2
x
,
=
,
b)
=
a)
x − y3
y0
2y − x2
y0
0 ax + by + f (x, y)
x
∂f ∂g ∂g
, where f, g, ∂f
=
c)
∂x , ∂y , ∂x , ∂y are zero at (0, 0).
cx + dy + g(x, y)
y0
• 8.1.101: Find the critical points and linearizations of the following systems.
a) x0 = sin πy + (x − 1)2 , y 0 = y 2 − y
b) x0 = x + y + y 2 , y 0 = x
c) x0 = (x − 1)2 + y, y 0 = y + x2 .
• 8.1.103: The concepts of critical points and linearization also generalize to higher
dimensions, by adding more functions and more variables to the Jacobian matrix. For
the following system of 3 equations, find the critical points and their linearizations:
 0  

x
x + z2
 y0  =  z2 − y  .
z0
z + x2
• 8.1.104: Write down the non-autonomous system of 2 equations x0 = f (x, y, t),
y 0 = g(x, y, t) (where x = x(t), y = y(t)) as an autonomous system of 3 equations,
using variables u, v, w.
• 8.2.1: For the systems below, find and classify the critical points (this includes a
discussion of stability).
0 0 2
0 x
−x + 3x2
x
x + y2 − 1
x
yex
a)
=
, b)
=
, c)
=
.
y0
−y
y0
x
y0
y − x + y2
• 8.2.3 (modified): Find the critical point of the following system, and determine
whether the system is almost-linear at it:
0 x
−x2
=
.
y0
−y 2
• 8.2.7: Consider the system x0 = f (x, y), y 0 = g(x, y). Suppose that g(x, y) > 1 for
all (x, y). Are there any critical points? What can we say about the trajectories as t
goes to infinity?
• 8.2.101: For the systems belows, find and classify the critical points.
0 0 0 xy
x
y − y2 + x
x
−x + x2
x
.
=
,
c)
=
,
b)
=
a)
x+y−1
y0
−x
y0
y
y0
2