MATH 2230 Assignment 5 Topic: Laplace transforms

MATH 2230 Assignment 5
Topic: Laplace transforms
1. Use the definition of the Laplace transform to determine
For which values of s is your answer valid?
L[t 2 ].
2.
a) Show that
by using the definition of the Laplace transform.
L[t 0 ] = 0!
s 0+1 when s > 0 (1)
b) Let n 1 be an integer. Prove that
∫
L[t n ∞
] = e −st t n dt
0
∫ )
R
= lim e
R→∞( −st t n dt
0
( [ ] ) e
= lim −
−st t n R ∫ )
+ n R
R→∞ s
0
s R→∞( lim e −st t n−1 dt
0
and hence show that for each n = 1, 2,
L[t n ]= n s L[tn−1 ] when s >0. (2)
c) Use induction to show that for each n =0, 1, 2,
L[t n ]= n!
s n+1 when s > 0.
(Hint: use part (a) for the n = 0 case and part (b) for the induction step).
3. Show that y(t) = t 3 is of exponential order.
4. Consider the following existence theorem:
Theorem 1. (Existence) Let y(t) be a function such that |y(t)|b.
Show that the function 3cos(2t) is of exponential order and use the above theorem to show that
exists for s > 0.
L[3cos(2t)]

5. Use any formulae done in class to determine the following
i. L[t 3 −t 2 + 4t +2]
ii. L[3e 4t − e −2t ]
iii. L[cos 2 2t]
iv. L[t 2 e 2t ]
v. L[φ(t)] where φ(t)=
(hint: use a double angle formula)
{
sin 2t 0 t < π
0 t π
6. Use partial fractions to determine the following
[ ]
i. L −1 s 2 − 6
s 3 + 4s 2 +3s
[ ]
ii. L −1 5s 3 −6s − 3
s 3 (s+1) 2
7. By shifting the s variable, determine the following
[ ]
i. L −1 1
s 2 + 4s + 4
[ ]
ii. L −1 2s+3
(s+4)
[
3 ]
iii. L −1 s 2
(s − 1) 4
8. By shifting the s variable, prove the following
[ ]
i. L −1 1
(s+a) n = tn e −at
for n = 0, 1, 2,
n!
[
ii. L −1 s
]= 1 (s+a) 2 + b 2 b e−at (b cos(bt) −asin(bt))
9. By shifting the t variable, find
and sketch your answer.
L −1 [ 3
s − 4e−s
s 2
]
+ 4e−3s
s 2
10. Using the Laplace transform, solve the differential equation
x ′′ + 3x ′ + 2x=4t 2
subject to the initial conditions x(0) = 0 and x ′ (0) = 0.
11. Using the Laplace transform, solve the differential equation
y ′′′′ − y = 0
subject to the initial conditions y(0)=0, y ′ (0)=1, y ′′ (0)=0 and y ′′′ (0)=0.