Page 256 - Schaum's Outline of Differential Equations
P. 256

CHAP.  23]             CONVOLUTIONS   AND THE UNIT  STEP  FUNCTION                   239



         23.16.  Prove that/(;t)  *  [g(x)  + h(x)]  =f(x)  * g(x)  +f(x)  *  h(x).













         23.17.  The following  equation is called an integral  equation  of  convolution  type.
               Assuming  that the Laplace Transform  for y(x)  exists,  we solve  this equation, and the next two examples,
               for  y(x).






                  We see that this integral equation  can be written as y(x)  =x  + y(x)  * sin x.  Taking the  Laplace transform  ££ of
               both  sides and applying Theorem  23.2,  we have




                        c
               Solving for £(y}  yields




                                   x 3
               This implies that  y(x)  = x-\  , which is indeed  the solution, as can be verified  by direct  substitution as follows:
                                    6






         23.18.  Use Laplace Transforms  to solve  the integral equation of convolution  type:






                  Here  we have y(x) = 2 -  y(x)  * e*. Continuing as in Problem  23.17, we find  that




               which  gives y(x)  = 2 -  2x as the desired  solution.

         23.19.  Use Laplace Transforms  to solve  the integral equation of convolution  type:






                                                                                     4
                                                   (
                                                                                              2
                                 3
                  Noting that y(x)  = x  + 4 * y(x),  we  find  that £{y} =  which  gives  y(x)  =  (-l + e *  -4x-8x )as
               the solution.
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