Note the following repetitive properties exhibited by the above table.
(i) The derivative property, when expressed in words states, Differentiation of func-
tion in the t-domain is represented in the s-domain by multiplication of the trans-
form function by sand subtracting the initial value of the function differentiated.
Note that the second derivative and higher derivatives follow this rule. For
example, if f′(t)and f′′(t)are the functions differentiated, then
L{ f′′(t)}= s[sF (s)−f(0 +)] −f′(0+)
︸ ︷︷ ︸
s times transform of
function differentiated
minus initial value
of function differentiated
L{ f′′′(t)}=︸s[s^2 F(s)−sf (0 +)︷︷−f′(0 +)] −f′′(0 +)︸
s times transform of
function differentiated
minus initial value
of function differentiated
(ii) Multiplication by tin the t-domain corresponds to a differentiation in the s-
domain multiplied by a -1.
(iii) Division by tin the t-domain corresponds to an integration from sto ∞in the
s-domain.
Example 12-16. Laplace transform
Use Laplace transform techniques to solve the differential equation dy
dt
=αy with
initial condition y(0) = 1 ,where αis a known constant.
Solution
Here y=y(t)is a function of time t and taking the Laplace transform of both
sides of the given differential equation produces
L
{
dy
dt
}
=L{ α y}
Let Y(s) = L{ y(t)}denote the transform in the s-domain and make note that
L{ y′(t)}=L
{
dy
dt
}
=sY (s)−y(0) and L{ α y (t)}=αL{ y(t)}=α Y (s)
