Victor Camocho
math2250fall2011-2
WeBWorK assignment number Homework 12 is due : 11/17/2011 at 11:00pm MST.
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Now find the inverse transform of Y (s) to find the solution
y(t).
1. (1 pt) hw12/p1.pg
Find the inverse Laplace transform of
y(t) =
−2e−3s + 1e−6s − 3e−8s + 4e−11s
F(s) =
s
f (t) =
.
.
6. (1 pt) hw12/p6.pg
Take the Laplace transform of the following initial value and
solve for Y (s) = L {y(t)}:
2. (1 pt) Library/274/Laplace3/prob4.pg
Find the Laplace transform of
f (t) = 2u4 (t) + 4u6 (t) − 4u8 (t)
(
sin(πt), 0 ≤ t < 1
y + 1y =
0,
1≤t
00
F(s) =
.
3. (1 pt) hw12/p3.pg
Find the inverse Laplace transform of
F(s) =
Y (s) =
.
e−6s
s2 + 3s − 10
f (t) =
Now find the inverse Laplace transform of Y (s) to find y(t).
.
y(t) =
4. (1 pt) hw12/p4.pg
Find the inverse Laplace transform of
F(s) =
6e−9s
s2 + 36
f (t) =
7. (1 pt) hw12/p7.pg
Take the Laplace transform of the following initial value problem and solve for Y (s) = L {y(t)}:
.
5. (1 pt) hw12/p5.pg
Take the Laplace transform of the following initial value problem and solve for Y (s) = L {y(t)}:
(
1, 0 ≤ t < 1
y − 3y − 40y =
0, 1 ≤ t
00
y(0) = 0, y0 (0) = 0
0
Y (s) =
y00 − 5y0 − 6y = S(t)
y(0) = 0, y0 (0) = 0
(
1, 0 ≤ t < 1
Where S(t) =
,
S(t + 2) = S(t).
0, 1 ≤ t < 2
.
Y (s) =
NOTE: S(t) is the a square wave function as depicted in the
y(0) = 0, y0 (0) = 0
.
1
graph below.
depicted in the graph below:
HINT: It may be more helpful to think of S(t) defined equivalently as follows:
(
1, t ∈ [2n, 2n + 1)
S(t) =
,
n∈N
0, t ∈ (2n + 1, 2n + 1]
10. (1 pt) Library/274/Laplace/prob13.pg
Find the inverse Laplace transform of
8. (1 pt) hw12/p8.pg
Use the Laplace transform to solve the following initial value
problem:
y(t) =
y00 − 4y0 − 45y = δ(t − 9)
9s2 + 8s + 3
(s2 − 16s + 73)(s2 + 9)
(
4t, 0 ≤ t ≤ 5
y + 64y =
20, t > 5
y(0) = 0, y0 (0) = 0
00
(3)
R∞
Using Y (s) for the Laplace transform of y(t), (i.e., Y (s) =
form of the differential equation and solving for Y (s)
Y (s) =
−∞ δ(t) dt
−∞
y(0) = 0, y0 (0) = 0
L {y(t)}), find the equation you get by taking the Laplace trans-
(1) δ(t) = 0 for all t 6= 0
R∞
.
11. (1 pt) hw12/p11.pg
Consider the following initial value problem:
y(t) =
.
NOTE: The function δ(t) is the Dirac Delta function and satisfies the following properties:
(2)
s>8
12. (1 pt) hw12/p12.pg
Use the Laplace transform to solve the following initial value
problem:
f (t)δ(t − a) dt = f (a)
9. (1 pt) hw12/p9.pg
Take the Laplace transform of the following initial value problem and solve for Y (s) = L {y(t)}:
y00 − 8y0 + 20y = 0
y(0) = 0, y0 (0) = 5
First, using Y (s) for the Laplace transform of y(t), (i.e.,
Y (s) = L {y(t)}), find the equation you get by taking the Laplace
transform of the differential equation and solve for Y (s)
y00 + 12y0 + 16y = T (t)
y(0) = 0, y0 (0) = 0
(
t,
0 ≤ t < 1/2
Where T (t) =
,
T (t + 1) = T (t).
1 − t, 1/2 ≤ t < 1
Y (s) =
.
Y (s) =
By completing the square in the denominator find the inverse
Laplace transform of Y (s) to find y(t)
y(t) =
NOTE: The function T (t) is a triangular wave function as
c
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.