219HW3.pdf

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Math 219, Homework 3
Due date: 9.12.2005, Friday
1.
Consider the initial value problem
d
2
x dx
+
+
x
=
u
4
(t),
dt
2
dt
y(0)
=
y
(0) = 0
(a)
Solve this initial value problem using the Laplace transform.
dx
dt
with respect to
t
(You can use the function
Step(t,
4) to create a unit step function
with discontinuity at
t
= 4).
(b)
Use ODE Architect to solve the equation, and graph the solution. Also graph
(c)
Discuss how the graphs agree with the solutions in (a): in particular determine
dx
(if any) all the points where
x(t)
and
are discontinuous, behavior of these two
dt
functions for
t
→ ∞,
their maxima and minima.
2.
Write each of the following systems of differential equations in matrix form, find
the eigenvalues and eigenvectors of the coefficient matrices, and using these, find
all solutions of each system. Also, graph the phase portraits (x
−
y
graph) using
ODE Architect. Please use a scale which includes the point (0, 0), and graph several
solutions in order to clearly observe the behavior around (0, 0). Also, place arrows
on the solution curves which indicate the direction of increasing
t,
and make sure
that solution curves along the eigenvector directions are graphed if there are any
real eigenvectors.
(a)
dx
= 2x
−
y
dt
dy
= 3x + 3y
dt
(b)
dx
=
−x
+
y
dt
dy
= 3x
−
4y
dt
(c)
dx
= 2x + 3y
dt
dy
= 5x + 5y
dt
(d)
dx
=
−4x
+ 3y
dt
dy
=
−3x
+ 2y
dt
(e)
dx
=
−x −
3y
dt
dy
= 2x +
y
dt

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