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这是一篇是完成拉普拉斯变换相关的数学问题
Math 123
Practice Final Exam
May 12, 2020
NAME:
(Please Print)
DIRECTIONS:
Do not open Exam until instructed to do so.
Do each of the problems and show all work.
NO WORK MEANS NO POINTS!
Box or circle and LABEL your final solution.
You will have 135 minutes to complete this Exam.
SCORES:
1. /20
2. /10
3. /15
4. /15
5. /20
6. /20
7. /10
8. /10
9. /10
Total: /130
1. 20 points Solve
(a) ˙x = x
2
t
(b) 2xy0 y = x for x > 0.
1
2. 10 points Use the properties of matrices to prove that multiplication by a m × n matrix A
is a linear transformation from R
n
to R
m.
To be more formal, let v ∈ R
n and LA(v) = Av. Explain why Av ∈ R
m and show LA(v)
satisfies the definition of a linear transformation.
2
3. 15 points Let A =

1 2 3
4 5 6
and B =

1 2
3 4
.
(a) Find AB.
(b) Find BA.
(c) Find B1
.
3
4. 15 points Let
A =

0 0 0 9 1
2 0 0 0 0
0 0 3 0 0
0 1 0 0 0
1 0 0 0 6

.
(a) Using any method, calculate the determinant of A, det(A).
(b) Is A invertible Why or why not
(c) How many solutions does Ax = 0 have Why
(d) Extra Credit: Calculate the adjoint (adjugate) of A.
(e) Extra Credit: Find the inverse of A.
4
5. 20 points Let
x¨ 2 ˙x + x = 0.
(a) Verify x1(t) = e
t and x2(t) = tet are solutions.
(b) Prove S = {x1, x2} is a linearly independent set of solutions.
(c) What is the dimension of the solution space
5
6. 20 points Let
A =

11 8
8 9
.
Find the solution to
˙

x
y

= A

x
y

with x(0) = 3 and y(0) = 4.
6
7. 10 points Let
x¨ x = e
t
with x(0) = ˙x(0) = 0. Find the impulse response and use a convolution to solve for x(t).
7
8. 10 points
˙

x
y

=

5 1
4 1 x
y

, P =

1 1
2 1
. Let
u
w

= P
1

x
y

=

1 1
2 1
x
y

=

x + y
2x y

.
(a) Prove
˙

u
w

= P
1
˙

x
y

=

3u + w
3w

=

3 1
0 3 u
w

.
(b) Solve the system in (a) for
u
w

.
(c) Use the answer in (b) to find
x
y

.
8
9. 10 points Two planes are given by
4x + 2y + 3z = 3 and 3x + 4y + 3z = 4
Find the line, P0 + αv with P0, v ∈ R
3 and α ∈ R, where the two planes intersect.
9
1 Laplace Transforms
f(t) L{f(t)}(s) f(t) L{f(t)}(s)
1
1
s
e
at 1
s + a
t
1
s
2
t
n n!
s
n+1
cos ω0t
s
s
2 + ω
2
0
sin ω0t
ω0
s
2 + ω
2
0
cosh ω0t
s
s
2 ω
2
0
sinh ω0t
ω0
s
2 ω
2
0
u(t a)
e
as
s
δ(t a) e
as
Table 1: Laplace Transforms
Theorem 1. Let y
0 + p(x)y = q(x) with p(x) and q(x) continuous. Then
y(x) = Ce
R x
x0
p(t) dt + e

R x
x0
p(t) dt Z x
x0
e
R t
x0
p(s) ds
q(t) dt
for any real number C.
10