Filter
Top Products
Page 16-10
Press J @ODETY to get the string “Linear w/ cst coeff” for the ODE type in
this case.
Laplace Transforms
The Laplace transform of a function f(t) produces a function F(s) in the image
domain that can be utilized to find the solution of a linear differential equation
involving f(t) through algebraic methods. The steps involved in this application
are three:
1. Use of the Laplace transform converts the linear ODE involving f(t) into an
algebraic equation.
2. The unknown F(s) is solved for in the image domain through algebraic
manipulation.
3. An inverse Laplace transform is used to convert the image function found in
step 2 into the solution to the differential equation f(t).
Definitions
The Laplace transform for function f(t) is the function F(s) defined as
The image variable s can be, and it generally is, a complex number.
Many practical applications of Laplace transforms involve an original function
f(t) where t represents time, e.g., control systems in electric or hydraulic circuits.
In most cases one is interested in the system response after time t>0, thus, the
definition of the Laplace transform, given above, involves an integration for
values of t larger than zero.
The inverse Laplace transform
maps the function F(s) onto the original function
f(t) in the time domain, i.e., L
-1
{F(s)} = f(t).
The convolution integral
or convolution product of two functions f(t) and g(t),
where g is shifted in time, is defined as
0
{()} () () .L
∞
−
== ⋅
∫
st
ft Fs ft e dt
.)()())(*(
0
∫
⋅−⋅=
t
duutguftgf