Module

for

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  [Graphics:Images/LaplaceShiftingMod_gr_1.gif]  and  [Graphics:Images/LaplaceShiftingMod_gr_2.gif].  Theorem 12.15 (the first shifting theorem) shows how their transforms are related to those of   [Graphics:Images/LaplaceShiftingMod_gr_3.gif]  and  [Graphics:Images/LaplaceShiftingMod_gr_4.gif]  by shifting the variable s in [Graphics:Images/LaplaceShiftingMod_gr_5.gif].  A companion result, called the second shifting theorem, Theorem 12.16, shows how the transform of [Graphics:Images/LaplaceShiftingMod_gr_6.gif] can be obtained by multiplying [Graphics:Images/LaplaceShiftingMod_gr_7.gif] by  [Graphics:Images/LaplaceShiftingMod_gr_8.gif].  Loosely speaking, these results show that multiplication of [Graphics:Images/LaplaceShiftingMod_gr_9.gif] by  [Graphics:Images/LaplaceShiftingMod_gr_10.gif]  corresponds to shifting [Graphics:Images/LaplaceShiftingMod_gr_11.gif],  and that shifting [Graphics:Images/LaplaceShiftingMod_gr_12.gif] corresponds to multiplication of the transform [Graphics:Images/LaplaceShiftingMod_gr_13.gif] by  [Graphics:Images/LaplaceShiftingMod_gr_14.gif].  

 

Theorem 12.15   (Shifting the Variable  s).  If  [Graphics:Images/LaplaceShiftingMod_gr_15.gif]  is the Laplace transform of  [Graphics:Images/LaplaceShiftingMod_gr_16.gif],  then  

                        [Graphics:Images/LaplaceShiftingMod_gr_17.gif].  

Proof.

 

Definition 12.3   (The Unit Step Function).  Let [Graphics:Images/LaplaceShiftingMod_gr_18.gif].  Then, the unit step function [Graphics:Images/LaplaceShiftingMod_gr_19.gif] is

                         [Graphics:Images/LaplaceShiftingMod_gr_20.gif]  

             [Graphics:Images/LaplaceShiftingMod_gr_21.gif]

            Figure 12.22.  The graph of the unit step function [Graphics:Images/LaplaceShiftingMod_gr_22.gif].

 

Theorem 12.16   (Shifting the Variable  t).  If [Graphics:Images/LaplaceShiftingMod_gr_23.gif] is the Laplace transform of [Graphics:Images/LaplaceShiftingMod_gr_24.gif], and [Graphics:Images/LaplaceShiftingMod_gr_25.gif], then  

                         [Graphics:Images/LaplaceShiftingMod_gr_26.gif],

where  [Graphics:Images/LaplaceShiftingMod_gr_27.gif] and  [Graphics:Images/LaplaceShiftingMod_gr_28.gif]  are illustrated in Figure 12.23.

[Graphics:Images/LaplaceShiftingMod_gr_29.gif] [Graphics:Images/LaplaceShiftingMod_gr_30.gif]

            Figure 12.23.  Comparison of the functions  [Graphics:Images/LaplaceShiftingMod_gr_31.gif]  and  [Graphics:Images/LaplaceShiftingMod_gr_32.gif].  

Proof.

 

Example 12.17.  Show that  [Graphics:Images/LaplaceShiftingMod_gr_33.gif].

Solution.

If we let [Graphics:Images/LaplaceShiftingMod_gr_34.gif], then [Graphics:Images/LaplaceShiftingMod_gr_35.gif], and if we apply Theorem 12.15, we obtain the desired result:

            [Graphics:Images/LaplaceShiftingMod_gr_36.gif].

Explore Solution 12.17.

 

Example 12.18.   Show that  [Graphics:Images/LaplaceShiftingMod_gr_46.gif].  

Solution.

If set [Graphics:Images/LaplaceShiftingMod_gr_47.gif], and then set [Graphics:Images/LaplaceShiftingMod_gr_48.gif].  We apply Theorem 12.16 to get

            [Graphics:Images/LaplaceShiftingMod_gr_49.gif]

Explore Solution 12.18.

 

Extra Example 1.  Use Theorem 12.16 and find  [Graphics:Images/LaplaceShiftingMod_gr_56.gif].  

Explore Solution for Extra Example 1.

 

Example 12.19.  Find  [Graphics:Images/LaplaceShiftingMod_gr_63.gif]  if  [Graphics:Images/LaplaceShiftingMod_gr_64.gif] is as given in Figure 12.24.

     [Graphics:Images/LaplaceShiftingMod_gr_65.gif]

                 Figure 12.24.  The function [Graphics:Images/LaplaceShiftingMod_gr_66.gif].

Solution.

We represent  [Graphics:Images/LaplaceShiftingMod_gr_67.gif] in terms of step functions  [Graphics:Images/LaplaceShiftingMod_gr_68.gif].  Using the result of Example 12.18 and linearity, we obtain

    [Graphics:Images/LaplaceShiftingMod_gr_69.gif]

Explore Solution 12.19.

 

Example 12.20.  Solve the initial value problem  

                             [Graphics:Images/LaplaceShiftingMod_gr_77.gif]    

                 [Graphics:Images/LaplaceShiftingMod_gr_78.gif]

                                A graph of the solution.

Solution.

As usual, we let [Graphics:Images/LaplaceShiftingMod_gr_79.gif] denote the Laplace transform of [Graphics:Images/LaplaceShiftingMod_gr_80.gif].  The right hand side of the D.E. is  [Graphics:Images/LaplaceShiftingMod_gr_81.gif] and   [Graphics:Images/LaplaceShiftingMod_gr_82.gif] .  
Taking Laplace transforms we write  [Graphics:Images/LaplaceShiftingMod_gr_83.gif].  Using the initial conditions [Graphics:Images/LaplaceShiftingMod_gr_84.gif], [Graphics:Images/LaplaceShiftingMod_gr_85.gif] and [Graphics:Images/LaplaceShiftingMod_gr_86.gif] we get

            [Graphics:Images/LaplaceShiftingMod_gr_87.gif].

Solving for [Graphics:Images/LaplaceShiftingMod_gr_88.gif] yields

            [Graphics:Images/LaplaceShiftingMod_gr_89.gif].

Use the facts that  [Graphics:Images/LaplaceShiftingMod_gr_90.gif],  [Graphics:Images/LaplaceShiftingMod_gr_91.gif] and get  [Graphics:Images/LaplaceShiftingMod_gr_92.gif], [Graphics:Images/LaplaceShiftingMod_gr_93.gif], respectively.   Then we will apply Theorem 12.16.  We compute the solution, [Graphics:Images/LaplaceShiftingMod_gr_94.gif], as

        [Graphics:Images/LaplaceShiftingMod_gr_95.gif]  

Then using the trigonometric identity [Graphics:Images/LaplaceShiftingMod_gr_96.gif] we can write this in a more familiar form   

                [Graphics:Images/LaplaceShiftingMod_gr_97.gif]

Explore Solution 12.20.

 

Exercises for Section 12.7.  The Laplace Transform: Shifting Theorems and the Step Function

 

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Laplace Transform

 

 

 

  

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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