MATH 2230 Assignment 5 Topic: Laplace transforms

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.