Math 2280-001
Mon Apr 13
7.1-7.4 Laplace transform, and application to DE IVPs, especially those in Chapter 3. Today we'll
continue to fill in the Laplace transform table (at the end of the notes), that we just started using on Friday.
Along the way we'll revisit some of the mechanical oscillation differential equations from Chapter 3.
Exercise 1) (to review) Use the table to compute
1a) L 4 C 2eK4 t s
2
6
1b) LK1
C
t .
sK2
s
Exercise 2) (to review the definition) Use the definition of Laplace transform,
N
F s =L f t
f t eKs t dt
s d
0
to find the Laplace transform of the step function graphed below. (The function is equal to zero for t R 3.)
2
1
0
0
1
2
t
3
4
5
Exercise 3a) Use the Table entry we proved on Friday for derivatives (via integration by parts), namely
L f# t s = s L f t s K f 0 = s F s K f 0
and math induction to show that for n 2 ;
L f n t s = sn F s K sn K 1f 0 K sn K 2f# 0 K...Ks f
nK2
0 Kf
nK1
0 .
3b) (Integrals are "negative" derivatives): Use the Laplace transform first-derivative formula above to
show that
t
1
F s
L
f t dt s = L f t s =
s
s
0
t
r
L
0
0
f t dt dr
s =
F s
s2
...
Exercise 4) Use the result of 3a to show that for n 2 ;,
L t s =
L t2
L tn
Exercise 5) Find LK1
1
2
s s C4
a) using the result of 3b.
b) using partial fractions.
t
s =
s =
1
s2
2
s3
n!
sn C 1
.
Exercise 6) (first translation theorem). Use the definition of Laplace transform to show that
L ea t f t s = L f t s K a = F s K a
Exercise 7) As a special case of Exercise 6, show
L t ea t
L tn ea t
s =
s =
1
sKa
n!
sKa
2
.
nC1
Exercise 8) Use the result of Exercise 7 for a = i k,
L t ei k t
s =
1
sKk i
2
to verify that
a) L t cos k t
s =
b) L t sin k t
s =
s2 K k2
s2 C k2
2ks
2
s2 C k2
2
.
Use those results and algebra to deduce these (other) inverse Laplace transform table entries related to
resonance phenomena:
c) LK1
d) LK1
s
2 2
s2 C k
1
2 2
s2 C k
t =
t =
1
t sin k t
2k
1
2 k3
sin k t K k t cos k t
.
Exercise 9) 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 )? How would you have tried to solve this problem in Chapter 3?
Exercise 10) Solve the following IVP. Use this example to recall the general partial fractions algorithm.
x## t C 4 x t = 8 t e2 t
x 0 =0
x# 0 = 1
How to check partial fractions with Maple:
8
1
> convert
C
, parfrac, s ;
s K 2 2 $ s2 C 4
s2 C 4
1
1
1 sC2
K
C
C
2
2 sK2
2 s2 C 4
sK2
>
(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
tf t
t2 f t
tn f t , n 2 Z
(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
2 C k2
(s O a
A
sOa
A
k
sKa
A
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
KF# s
F## s
K1 n F n s
A
A
f t
t
t cos k t
1
2 k t sin k t
1
2 k3
sin k t K k t cos k t
N
F s ds
s
s2 K k2
s2 C k2
s
2
2
s2 C k2
1
2
s2 C k2
eat f t
t ea t
tn e a t, n 2 Z
F sKa
1
sKa
n!
sKa
Laplace transform table
2
nC1