Chapter 12  Fourier Series and the Laplace Transform

12.8  Laplace Transforms: Multiplication and Division by t

    This section is a continuation of our development of the Laplace Transforms in Section 12.5, Section 12.6 and Section 12.7.

    Sometimes the solutions to nonhomogeneous linear differential equations with constant coefficients involve the functions  , , or    as part of the solution.  We now show how the Laplace transforms of    and    are related to the Laplace transform of .  The transform of    will be obtained via differentiation and the transform of    will be obtained via integration.  To be precise, we present Theorems 12.17 and 12.18.

Theorem 12.17   (Multiplication by t).  If    is the Laplace transform of  ,  then

                        .  

Proof.

Theorem 12.18  (Division by t).   Let both    have Laplace transforms and let    denote the Laplace transform of  .   If    exists then  

                        .  

Proof.

Example 12.21.  Show that  .  

Solution.

If we let , then .  Hence we can differentiate to obtain the desired result:

            

Explore Solution 12.21.

Example 12.22.  Show that  .

Solution.

We let , then .  Because ,  we can integrate to obtain the desired result:  

            

Explore Solution 12.22.

    Some types of differential equations involve the terms    or  .  We can use Laplace transforms to find the solution if we use the additional substitutions

(12.32)            ,  
                    and
(12.33)            .

Example 12.23.  Use Laplace transforms to solve the initial value problem

                

            Some of the functions in the family of solutions.

Solution.

If we let denote the Laplace transform of and substitute Equations (12.32) and (12.33) into the preceding equation, we get

,  which can be simplifies as    and then rewritten in the form  

(12.34)              

Equation (12.34) involves and can be written as a first-order linear differential equation

(12.35)            .  

The integrating factor  for the differential equation is  

            .

Multiplying Equation (12.35) by  produces  

            ,

which in turn can be written as  

            .

Now integrate both sides and obtain       which yields

             ,

where C is the constant of integration.  Hence the solution to Equation (12.35) is

            .

Thus   in this equation is the desired solution:

            .

Remark.  For this differential equation there is a family of solutions.

Explore Solution 12.23.