Alyssa Kritz Milliken MAT 275 Fall 2019
Assignment Section 6.4 Discontinuous Forcing Functions due 11/12/2019 at 11:59pm MST
1. (1 point)
Consider the following initial value problem:
y00 +49y=(5,0≤t≤7
0,t>7y(0) = 4,y0(0) = 0
Using Yfor the Laplace transform of y(t), i.e., Y=L{y(t)},
find the equation you get by taking the Laplace transform of the
differential equation and solve for
Y(s) =
2. (1 point)
Take the Laplace transform of the following initial value
problem and solve for Y(s) = L{y(t)}:
y00 −7y0−8y=(1,0≤t<1
0,1≤ty(0) = 0,y0(0) = 0
Y(s) = .
Now find the inverse transform:
y(t) = .
(Notation: write u(t-c) for the Heaviside step function uc(t)
with step at t=c.)
Note:
1
s(s−8)(s+1)=−1
8
s+
1
9
s+1+
1
72
s−8
3. (1 point)
Take the Laplace transform of the following initial value and
solve for Y(s) = L{y(t)}:
y00 +y=(sin(πt),0≤t<1
0,1≤ty(0) = 0,y0(0) = 0
Y(s) = . Hint: write the right hand
side in terms of the Heaviside function.
Now find the inverse transform:
y(t) = .
Note:
π
(s2+π2)(s2+1)=π
π2−11
s2+1−1
s2+π2
(Notation: write u(t-c) for the Heaviside step function uc(t)
with step at t=c.)
4. (1 point)
Consider the following initial value problem:
y00 +49y=(4t,0≤t<7
0,t≥7y(0) = 0,y0(0) = 0
Using Yfor the Laplace transform of y(t), i.e., Y=L{y(t)},
find the equation you get by taking the Laplace transform of the
differential equation and solve for
Y(s) =
5. (1 point)
Consider the following initial value problem:
y00 +64y=(8t,0≤t≤5
40,t>5y(0) = 0,y0(0) = 0
Using Yfor the Laplace transform of y(t), i.e., Y=L{y(t)},
find the equation you get by taking the Laplace transform of the
differential equation and solve for
Y(s) =
6. (1 point)
Take the Laplace transform of the following initial value
problem and solve for Y(s) = L{y(t)}:
y00 −2y0−35y=S(t)y(0) = 0,y0(0) = 0
where Sis a periodic function defined by
S(t) = (1,0≤t<1
0,1≤t<2,and S(t+2) = S(t)for all t≥0.
Hint: : Use the formula for the Laplace transform of a peri-
odic function.
1
Y(s) = .
The graph of S(t)(a square wave function):
7. (1 point)
Take the Laplace transform of the following initial value
problem and solve for Y(s) = L{y(t)}:
y00 +12y0+18y=T(t)y(0) = 0,y0(0) = 0
where Tis a periodic function defined by
T(t) = (t,0≤t<1/2
1−t,1/2≤t<1,and T(t+1) = T(t)for all t≥
0.
Y(s) = .
Graph of T(t)(a triangular wave function):
2
