# Math 215 Spring 2010 Assignment 9

Math 215
Spring 2010
Assignment 9
You are highly recommended to own a textbook, Boyce-DiPrima or another, to see a comprehensive treatment of the subjects and more examples.
&sect;9.1: 8(ab), 9(ab), 15, &sect;9.2: 4, 22(a), &sect;9.3: 8(abc), 9(abc)
&micro;
&para;
−1 −1
0
• (&sect;9.1: 8(ab)) For x =
x, (a) find the eigenvalues and eigenvectors,
0 −1/4
(b) classify the critical point (0, 0) as to type and determine whether it is stable,
asymptotically stable, or unstable.
&micro;
&para;
3 −4
• (&sect;9.1: 9(ab)) For x0 =
x, (a) find the eigenvalues and eigenvectors, (b)
1 −1
classify the critical point (0, 0) as to type and determine whether it is stable, asymptotically stable, or unstable.
&micro;
&para;
&micro; &para;
−1 −1
−1
0
• (&sect;9.1: 15) For x =
x+
, determine the critical point x = x0 ,
2 −1
5
and then classify its type and examine its stability by making the transformation
x = x0 + u.
• (&sect;9.2: 4) Sketch the trajectory corresponding to the solution satisfying the specified
initial conditions, and indicate the direction of motion for increasing t:
√
dx/dt = ay, dy/dt = −bx, a &gt; 0, b &gt; 0;
x(0) = a, y(0) = 0.
• (&sect;9.2: 22(a)) Find an equation of the form H(x, y) = c satisfied by the trajectory of
dx/dt = 2x2 y − 3x2 − 4y,
dy/dt = −2xy 2 + 6xy.
• (&sect;9.3: 8(abc)) Consider
dx/dt = x − x2 − xy,
1
dy/dt = (2y − y 2 − 3xy).
4
(a) Determine all critical points.
(b) Find the corresponding linear system near each critical point.
(c) Find the eigenvalues of each linear system. What conclusions can you then draw