Test 4,Math 3107 Name: _________________________
Spring 2000, Dr. Howard
Instructions: This take-home test is due no later than the beginning of class on Monday,
April 24. You must
work alone
on these problems. You may use your own text, your own
notes, your own calculator, and Maple, but you may use no other aids without my specific
permission. Please show all work and justify all answers.Staple your work with this sh
e
1.Consider the system of differential equations
Y
๎
๎
09
๎
16 Y
a.Find the general solution Y
๎
t
๎
.
b.Sketch the configuration of solutions in the phase plane, indicating any
straight-line solutions.
c.Y
๎
t
๎
๎
๎
0,0
๎
is an equilibrium solution of this system.
i.Identify it as a node, a saddle point, a center, or a spiral point, and
ii.determine if it is unstable, stable, or asymptotically stable.
2.Consider the system of differential equations
Y
๎
๎
715
๎
6
๎
11 Y
a.Find the general solution Y
๎
t
๎
.
b.Sketch the configuration of solutions in the phase plane, indicating any
straight-line solutions.
c.Y
๎
t
๎
๎
๎
0,0
๎
is an equilibrium solution of this system.
i.Identify it as a node, a saddle point, a center, or a spiral point, and
ii.determine if it is unstable, stable, or asymptotically stable.
d.Use Maple to draw the time series solutions, phase plane graph, and space curve
corresponding to the initial condition Y
๎
0
๎
๎
๎
1,1
๎
.
3.Compute eAt if Ais the matrix A
๎
๎
2
๎
5
47 .
4.Compute the Laplace transform of the function f
๎
t
๎
๎
e
๎
tif 0
๎
t
๎
5
0if t
๎
5.
1
