MATH 2090 : Fall 2011 Exam 4
Math 2090 Test 4 Britt Spring 2011
Work
on
the fronts
of
your paper.
Show
all work.
1.
Consider
the
system
a) Determine A, the matrix
of
coefficients for this linear system.
b)
Determine the eigenvalues and the associated eigenvectors for A.
c) Determine the general solution to
the
system.
d)
Solve the system subject
to
the
initial condition
x(O)
=
(~J
2.
Convert
the
following equation to a first order linear
system
...
y'"
+ty"
+3costy'
+4y
e
l
3. Determine
the
linear independence or dependence
of
the following vector functions.
You
must
prove your answer.
A)
(t
)=
[~et
]
x,
=
[:~,
]
4.
Assume
that/is
of
exponential order
on
[0,00)
, that J' exists and is continuous on
[0,00)
then
prove
that L(J') = sL(J) -J
(0).
Hint: Integration
by
parts.
5. Determine the Laplace transform
of
the functions below
A)
J(t)
e
7t
sin2t
B)
J(t)=u
3
(t)cos(t-3)
C) J
(t
) =
4t
3 +7e
f
-5sint
6. Determine
the
following inverse transforms
3s
-1
[
2e
l
C)
A) L 2 J
s
+4
7.
Solve the following initial value problem
by
using
Laplace Transforms
y" + y =e
2f
,
yeO)
=0 , y'(O) = 1
Please turn
over
Document Summary
You must prove your answer. (t ) =[~et] x, =[:~, : assume that/is of exponential order on [0,00) , that j" exists and is continuous on [0,00) then prove that l(j") = sl(j) - j (0). Hint: integration by parts: determine the laplace transform of the functions below, j(t) e7t sin2t, j(t)=u 3 (t)cos(t-3, j (t ) = 4t 3 + 7e f. 5 sin t: determine the following inverse transforms. C: solve the following initial value problem by using laplace transforms y" + y = e 2f. , yeo) = 0 , y"(o) = 1. L if i " = [0 2) i and ( ~i) are the associated eigenvectors of the matrix of coeffici ents. 2 0 then find the real solution to this system. Hint: 2i are eigenvalues: calculate the laplace transfonn of the periodic function l
