MAT 275 CHAPTER 6, 7 PRACTICE PROBLEMS
(Material from earlier sections are on previous reviews)
Given Laplace Transform Table:
6.3. Step Functions
1. Find the Laplace transform of the following functions.
(a) 𝑓(𝑡 𝑡
)=( )
𝑡 + 3 𝑢7( )
(b) 𝑓(𝑡)= 𝑡 (𝑡)
2𝑢3
(c) 𝑓(𝑡)={1, 0 ≤ 𝑡 < 2
𝑡2− 4𝑡 + 4, 𝑡 ≥ 2
(d) 𝑓(𝑡)={𝑡, 0 ≤ 𝑡 < 3
5, 𝑡 ≥ 3
(e) 𝑓(𝑡)={
0,
𝑡 − 𝜋,
0, 𝑡 < 𝜋
𝜋 ≤ 𝑡 < 2𝜋
𝑡 ≥ 2𝜋
(f) 𝑓(𝑡)={cos(𝜋𝑡) , 𝑡 < 4
0, 𝑡 ≥ 4
(g) 𝑓(𝑡)={𝑡, 0 ≤ 𝑡 < 1
𝑒𝑡, 𝑡 ≥ 1
2. Find the inverse Laplace transform:
(a) 𝐹(𝑠) = 𝑒−3𝑠
𝑠−2
(b) 𝐹(𝑠) = 1+𝑒−2𝑠
𝑠2+6
(c) 𝐹(𝑠)=3
𝑠+4
𝑠2+5𝑠
𝑠2+9 − 𝑒−3𝑠 (3
𝑠+4
𝑠2+5𝑠
𝑠2+9)
6.4. Solutions of IVP with Discontinuous Forcing Functions
𝑓(𝑡) = 𝐹(𝑠) ℒ{ }
−1
𝐹(𝑠)= ℒ{𝑓(𝑡)}
𝑦(𝑡)
𝑌(𝑠)
1
1
1
𝑠
2
𝑡𝑛
𝑛!
𝑠𝑛+1
3
𝑒𝑎𝑡
1
𝑠−𝑎
4
cos(𝑏𝑡)
𝑠
𝑠2+𝑏2
5
sin(𝑏𝑡)
𝑏
𝑠2+𝑏2
6
𝑒𝑎𝑡 cos(𝑏𝑡)
𝑠−𝑎
( )
𝑠−𝑎 2+𝑏2
𝑓(𝑡) = 𝐹(𝑠) ℒ{ }
−1
𝐹(𝑠)= ℒ{𝑓(𝑡)}
7
𝑒𝑎𝑡 sin(𝑏𝑡)
𝑏
( )
𝑠−𝑎 2+𝑏2
8
𝑢𝑐(𝑡)= 𝑢(𝑡 − 𝑐)
𝑒−𝑐𝑠
𝑠
9
𝑢𝑐(𝑡 𝑡
)𝑓( )
𝑒−𝑐𝑠ℒ{𝑓(𝑡 + 𝑐)}
10
𝑢𝑐(𝑡)𝑓( )
𝑡 − 𝑐
𝑒−𝑐𝑠𝐹(𝑠)
11
𝛿(𝑡 − 𝑐)
𝑒−𝑐𝑠
12
𝑦′(𝑡)
𝑠𝑌(𝑠)− 𝑦(0)
13
