Winter 2014 Math 377
Homework VII Solutions
For Problems 1 to 3, use Laplace transforms to solve the initial value
problem.
1.
u + 5u = h2
u(0) = 1
Transforming h2 uses the rule that L[ha (t)u(t a)] = U eat where we
use the function u(t) = 1 which has u(t
Winter 2014 Math 377
Homework III
For each matrix M , do the following for the system x = M x.
a. Find all the eigenvalues and associated eigenvectors (you may use Mathematica for this).
b. Graph any straight line solutions in the phase plane and add arro
Winter 2014 Math 377
Homework VII
For Problems 1 to 3, use Laplace transforms to solve the initial value
problem.
1.
u + u = sin(2t)
u(0) = 0
2.
u + u = h3 cos(t 3)
u(0) = 1
3.
u + 4u = 3h5 sin(t 5)
u(0) = 1, u (0) = 0
4.
u + 5u = h2
u(0) = 1
5. Determine
Winter 2014 Math 377
Homework V
We will reexamine the systems from Homework I. For this homework you
may not assume that x, y 0.
Consider the following two systems:
System A :
dx
dt
dy
dt
= 5x + 2xy
= 4y + 3xy
System B :
dx
dt
dy
dt
= 6x x2 4xy
= 5y 2xy 2
Winter 2014 Math 377
Homework VI
1. Use the denition of the Laplace transform to nd L[cos(t)].
2. Use the fact that
d2
dt2
3. Use the fact that
d
dt
cos(t) = 2 cos(t) to nd L[cos(t)].
cos(t) = sin(t) to nd L[sin(t)].
4. Find L[t2 e3t ], try to do it with
Winter 2014 Math 377
Homework I
We have briey examined a predator-prey system in which the population of
one species (prey) is harmed by the presence of another species (predator). A
cooperative system is one in which the population of both species is ben
Winter 2014 Math 377
Homework II
This is all linear algebra material. Some (all) of this material should have
been included in your Linear Algebra course (although it may have looked
dierent). If some of this was missing, now is your chance to learn it. I
Winter 2014 Math 377
Laplace Transforms
L[u ] = sU u0
L[c1 u1 + c2 u2 ] = c1 U1 + c2 U2
L[ueat ] = U (s a)
L[ha u(t a)] = U eas
L[u v] = U V
L[tn ] =
n!
sn+1
1
L[eat ] =
sa
L[sin(t)] = 2
s + 2
s
L[cos(t)] = 2
s + 2
L[a ] = eas
t
(f
f ( )g(t ) d
g)(t) =
0
Winter 2014 Math 377
Inclass I
For each system of the form x = M x do the following:
a. Determine the eigenvalues and associated eigenvectors.
b. Use the eigenvectors to determine solutions with linear curves in the phase
plane.
c. Use the eigenvalues to
Winter 2014 Math 377
Exam I Solutions
Be sure to check the point value of each problem and spend time accordingly.
I am less interested in the arithmetic calculations required to get the answer
than in the steps you follow. You may want to sketch out the
Winter 2014 Math 377
Homework II Solutions
I did most of these calculations in Mathematica, you can nd that notebook
on the website as well. Included are the tests I used to verify that I had done
the calculations correctly.
II.1 Verify that the vectors v
Winter 2014 Math 377
Homework VI Solutions
1. Use the denition of the Laplace transform to nd L[cos(t)]. We will
use L for the indenite integral and nd L rst.
L=
cos(t)est dt
u = cos(t)
dv = est dt
du = sin(t) dt v = 1 est
s
1
L = cos(t)est
sin(t)est dt
Winter 2014 Math 377
Inclass I
For each system of the form x = M x do the following:
a. Determine the eigenvalues and associated eigenvectors.
b. Use the eigenvectors to determine solutions with linear curves in the phase
plane.
c. Use the eigenvalues to
Winter 2014 Math 377
Exam I
Be sure to check the point value of each problem and spend time accordingly.
I am less interested in the arithmetic calculations required to get the answer
than in the steps you follow. You may want to sketch out the solution s
Winter 2014 Math 377
Exam II
1. (10 points) Use the denition of the Laplace transform to nd the
transform of u(t) = e4t h2 (t).
2. (16 points) Compute the Laplace transform of the following (feel free to
use the formulas):
a. (8 points) t2 e4t
b. (8 point
Winter 2014 Math 377
Homework V Solutions
We will reexamine the systems from Homework I. For this homework you
may not assume that x, y 0.
Consider the following two systems:
System A :
dx
dt
dy
dt
= 5x + 2xy
= 4y + 3xy
System B :
dx
dt
dy
dt
= 6x x2 4xy
Winter 2014 Math 377
Homework IV
For each matrix below, determine the general real solution of x = M x
and the particular solution with x(0) = (1, 1)T .
IV.1 M =
2 2
4 4
1 = 3 + i 7
v1 =
2 = 3 i 7
v2 =
1+i 7
4
1+i 7
4
1i 7
4
x1 (t) = e
=e
cos( 7t) + i sin
Winter 2014 Math 377
Exam II
1. (10 points) Use the denition of the Laplace transform to nd the
transform of u(t) = e4t h2 (t).
e4t h2 (t)est dt =
L [u] =
0
1
=
e(4s)t
4 s
e(4s)t dt
2
=0
2
1
1 2(s+4)
e(4s)2 =
e
4 s
s+4
If you tried to use the formulas, yo
Winter 2014 Math 377
Homework I
We have briey examined a predator-prey system in which the population of
one species (prey) is harmed by the presence of another species (predator). A
cooperative system is one in which the population of both species is ben
Winter 2014 Math 377
Homework IV
For each matrix below, determine the general real solution of x = M x
and the particular solution with x(0) = (1, 1)T .
IV.1 M =
2 2
4 4
IV.2 M =
3 1
4 1
IV.3 M =
2 1
4 2
Determine a matrix M so that the phase plane of x =