Shifting Theorems and the Step Function
Chapter 12 Fourier Series and the Laplace Transform
12.7 Laplace Transform Shifting Theorems and the Step Function
This section is a continuation of our development of the Laplace Transforms in Section 12.5 and Section 12.6.
We have shown how to use the Laplace transform to solve linear differential equations. Familiar functions that arise in solutions to differential equations are and . Theorem 12.15 (the first shifting theorem) shows how their transforms are related to those of and by shifting the variable s in . A companion result, called the second shifting theorem, Theorem 12.16, shows how the transform of can be obtained by multiplying by . Loosely speaking, these results show that multiplication of by corresponds to shifting , and that shifting corresponds to multiplication of the transform by .
Theorem 12.15 (Shifting
Variable s). If is
the Laplace transform of , then
the unit step function
Figure 12.22. The graph of the unit step function .
Theorem 12.16 (Shifting
the Variable t). If
is the Laplace transform of ,
where and are illustrated in Figure 12.23.
Figure 12.23. Comparison of the functions and .
Example 12.17. Show that .
If we let ,
and if we apply Theorem 12.15, we obtain the desired result:
Explore Solution 12.17.
Example 12.18. Show that .
If set ,
and then set . We
apply Theorem 12.16 to get
Explore Solution 12.18.
Extra Example 1. Use Theorem 12.16 and find .
Explore Solution for Extra Example 1.
Example 12.19. Find if is as given in Figure 12.24.
Figure 12.24. The function .
in terms of step functions . Using
the result of Example 12.18 and linearity, we obtain
Explore Solution 12.19.
12.20. Solve the initial value
A graph of the solution.
As usual, we let
denote the Laplace transform of . The
right hand side of the D.E. is
Taking Laplace transforms we write . Using the initial conditions , and we get
Use the facts that ,
and get ,
respectively. Then we will apply Theorem
12.16. We compute the solution, ,
Then using the trigonometric identity we can write this in a more familiar form
Explore Solution 12.20.
Exercises for Section 12.7. The Laplace Transform: Shifting Theorems and the Step Function
The Next Module is
Multiplication and Division by t
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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell