Math 2250-4
Mon Nov 11
Homework office hours Tuesday TBA, for HW due Wednesday.
Go over last spring exam 2, Wednesday or Thursday 4:30-6:00 Room TBA.
10.1-10.3 Laplace transform, and application to DE IVPs, especially those in Chapter 5. Today we'll
continue to fill in the Laplace transform table (on page 2), and to use the table entries to solve linear
differential equations.
Exercise 1) (to review) Use the table to compute
1a) L 4 K 5 cos 3 t C 2eK4 t sin 12 t s
2
1
1b) LK1
C 2
t .
sK2
s C2 sC5
Exercise 2a) (to review) Use Laplace transforms to solve the IVP we didn't get to on Friday, for an
underdamped, unforced oscillator DE. Compare to Chapter 5 method.
x## t C 6 x# t C 34 x t = 0
x 0 =3
x# 0 = 1
f t , with
f t
% CeM t
c f t Cc f t
1 1
2 2
N
f t eKs t dt for s O M
Fs d
0
Y
verified
c F s Cc F s
1 1
1
2 2
1
s
t
A
(s O 0
1
t2
s2
2
tn, n 2 ;
s3
n!
A
sn C 1
1
ea t
(s O R a
sKa
s
cos k t
sin k t
cosh k t
sinh k t
ea tcos k t
ea tsin k t
f
n
f# t
f ## t
t , n2;
t
f t dt
0
(s O 0
A
(s O 0
A
s2 C k2
k
s2 C k2
s
s2 K k2
k
s2 K k2
sKa
sKa
(s O k
(s O k
(s O a
2 C k2
sOa
2 C k2
s F s Kf 0
s K s f 0 K f# 0
sn F s K sn K 1f 0 K...Kf n K 1 0
Fs
s
s2F
tf t
t2 f t
tn f t , n 2 Z
f t
t
KF# s
F## s
K1 n F n s
t cos k t
s2 K k2
1
2 k t sin k t
N
F s ds
s
s2 C k2
s
s2 C k2
A
A
k
sKa
A
2
2
A
A
1
2 k3
1
sin k t K k t cos k t
2
s2 C k2
eat f t
F sKa
t ea t
1
sKa
n!
tn e a t, n 2 Z
sKa
2
nC1
more after exam!
Laplace transform table
work down the table ...
3a) L cosh k t
s =
3b) L sinh k t
s =
s
2
s K k2
k
s2 K k2
.
Exercise 4) Recall we used integration by parts on Friday to derive
L f# t s = sL f t s K f 0 .
Use that identity to show
a) L f## t s = s2 F s K s f 0 K f# 0 ,
b) L f### t s = s3 F s K s2 f 0 K s f# 0 K f## 0 ,
c) L f n t s = sn F s K sn K 1 f 0 K sn K 2 f# 0 K ... Kf n K 1 0 , n 2 ; .
t
F s
d) L
f t dt s =
.
s
0
These are the identities that make Laplace transform work so well for initial value problems such as we
studied in Chapter 5.
Exercise 5) Find LK1
1
t
2
s s C4
a) using the result of 4d.
b) using partial fractions.
Exercise 6) Show L tn
s =
n!
s
nC1
, n 2 ; , using the results of 4.
Tomorrow we will finish filling in the Laplace transform table entries. Notice how the Laplace transform
table is set up to use partial fraction decompositions for X s and linearity in order to find the inverse
Laplace transforms x t . And be amazed at how the table lets you quickly deduce the solutions to
important DE IVPs, like we studied in Chapter 5.
Exercise 7)
a) Use Laplace transforms to write down the solution to
2
x## t C w0 x t = F0 sin w0 t
x 0 = x0
x# 0 = v0 .
what phenomena do solutions to this DE illustrate (even though we're forcing with sin w0 t rather than
cos w0 t )?
b) Use Laplace transforms to derive the IVP solution below, in the case w s w0 .
2
x## t C w0 x t = F0 sin w t
x 0 = x0
x# 0 = v0 .