Laplace Transform
Laplace Transform
The Laplace transform is a very powerful tool for studying differential
equations. The control problems concern dynamic analogue
systems. Such situations are aptly described by differential
equations.
The Laplace transform helps us
To solve the homogenous equations and particular integral in one
operation
To convert differential equations to algebraic equations, solve such
simultaneous equations with ease and then get the final solution
through inverse Laplace Transform.
Laplace Transform
Let 𝑓 𝑡 be a function of t specified 𝑡 > 0. Then the
Laplace Transform of 𝑓(𝑡) , denoted by ℒ 𝑓 𝑡 is defined
as,
∞
ℒ𝒇 𝒕 = 𝑭(𝒔) = 𝒆 𝟎−𝒔𝒕 𝒇 𝒕 𝒅𝒕
Also write as,
𝑭 𝒔 = ℒ {𝒇 𝒕 }
Common Laplace Transform Formulas
𝑓(𝑡) 𝐹(𝑠)
Unit step function 𝝁(𝒕) 𝟏/𝒔
Exponential 𝒆𝒂𝒕 𝟏
𝒔−𝒂
𝒏𝒕𝒉 power 𝒏!
𝒔𝒏+𝟏
𝐬𝐢𝐧(𝝎𝒕) 𝝎
𝒔𝟐 + 𝝎𝟐
𝐜𝐨𝐬(𝝎𝒕) 𝒔
𝒔𝟐 + 𝝎𝟐
𝒔𝒊𝒏𝒉(𝝎𝒕) 𝝎
𝒔𝟐 − 𝝎𝟐
𝒄𝒐𝒔𝒉(𝝎𝒕) 𝒔
𝒔𝟐 − 𝝎𝟐
𝝁 𝒕−𝒂 𝒆−𝒂𝒔
𝒔
Linearity Property. If a and b are constants while 𝑓 𝑡 and 𝑔(𝑡) are functions
of 𝑡 whose Laplace transform exists, then
ℒ 𝒂𝒇 𝒕 + 𝒃𝒈(𝒕) = 𝒂ℒ 𝒇(𝒕) + 𝒃ℒ 𝒈(𝒕)
Example:
Find the Laplace Transform of 3𝑡 2 − 2𝑐𝑜𝑠2𝑡 + 4𝑒 −𝑡
Frequency Shifting Property of Laplace Transforms.
ℒ 𝒆±𝒂𝒕 𝒇(𝒕) = 𝑭 𝒔 ± 𝒂
where 𝐹 𝑠 is the Laplace transform of 𝑓(𝑡) and 𝑎 is a constant.
Examples: Find the Laplace Transform of the following function of time:
1.) ℒ 𝑒 −𝑡 𝑐𝑜𝑠2𝑡
2.) ℒ 𝑒 4𝑡 𝑒 −2𝑡 𝑠𝑖𝑛(5𝑡)
Time Shifting Property/ Laplace Transforms of Piecewise Continuous Function
Unit Step Function
The unit step function at time 𝑡, 𝑢 𝑡 , is defined as
0𝑡 <0
𝑢 𝑡 =ቊ
1𝑡 >0
Shifted Unit Step Function
A shifted unit step function has value of 0 up to the time 𝑡 = 𝑎 and has value 1 afterward.
0 𝑖𝑓 𝑡 < 𝑎
𝑢 𝑡−𝑎 =ቊ
1 𝑖𝑓 𝑡 > 𝑎
General Formula for Laplace Transform of Piecewise Functions:
Laplace Transform of a Step-Modulated Function.
Let g(t) defined on 0, ∞ , suppose 𝑎 ≥ 0, and assume ℒ 𝒈(𝒕 + 𝒂) exists for 𝑠 >
𝑠𝑜 . Then,
ℒ 𝒖 𝒕 − 𝒂 𝒈(𝒕) = 𝒆−𝒂𝒔 ℒ 𝒈(𝒕 + 𝒂)
Example:
1.) Find the Laplace Transform of 𝑓 𝑡 =
0 𝑖𝑓 𝑡 < 3
2.) Find the Laplace Transform of 𝑓 𝑡 = ൞2 𝑡 − 3 𝑖𝑓 3 ≤ 𝑡 < 7
8 𝑖𝑓 𝑡 > 7
Differentiation Property of Laplace Transform. (Theorem Multiplication by 𝒕𝒏 )
If 𝑓(𝑡) is a function of whose Laplace Transform is F(S), then the Laplace transform
of 𝑡 𝑛 𝑓 𝑡 is given by:
𝒏
ℒ 𝒕𝒏 𝒇(𝒕) = −𝟏 𝒏 𝒅 𝑭 𝒔 , 𝒏 = 𝟏, 𝟐, 𝟑
𝒅𝒔𝒏
Examples:
Find the Laplace Transform of the following equations:
1.) ℒ 𝑡𝑠𝑖𝑛𝑎𝑡
2.) ℒ 𝑡 2 𝑒 2𝑡
Laplace Transform of Derivatives
For a function 𝑓(𝑡), if 𝐹 𝑠 = ℒ 𝑓(𝑡) , then the Laplace transform of the derivative of
𝑓 𝑡 is:
First Derivative:
ℒ 𝒇′(𝒕) = 𝒔ℒ 𝒇(𝒕) − 𝐟(𝟎)
First Derivative:
ℒ 𝑓′(𝑡) = 𝑠F(𝑠) − f(0)
Second Derivative:
ℒ 𝑓′′(𝑡) = 𝑠 2 𝐹(𝑠) − 𝑠𝑓 0 + 𝑓′(0)
n-th Derivative
𝑑𝑛
ℒ 𝑛
𝑓(𝑡) = 𝑠 𝑛 𝐹 𝑠 − 𝑠 𝑛−1 𝑓 0 − 𝑠 𝑛−2 𝑓 ′ 0 . … _ − 𝑓 𝑛−1
(0)
𝑑𝑡
Examples:
1.) Find the Laplace transform of 2𝑦 ′ + 𝑦 = 𝑐𝑜𝑠𝑥 assuming that the initial
condition is zero.
𝑑2 𝑥 𝑑𝑥
2.) For the differential equation, + 6 𝑑𝑡 + 8𝑥 = 0 with the initial condition
𝑑𝑡 2
𝑥 0 = 1.
Additional Examples:
Find the Laplace Transform of the ff:
1.) 𝑦" + 2𝑦′ + 2𝑦 = 0 , 𝑦(0) = −3, 𝑦′(0) = 1
2.) 𝑒 7𝑡 𝑐𝑜𝑠ℎ2𝑡
3.) 𝑡 2 𝑒 3𝑡
4.) 𝑒 −2𝑡 + 5𝑡 𝑢 𝑡 − 2 − 𝑢(𝑡)
5.) 5𝑒 −3𝑡 + 𝑢 𝑡 − 1 − 𝑢(𝑡 − 2)
Properties of Laplace Transform
