Example 12.32.   Solve the initial value problem

              
            with  
            .  

                              Figure 12.31.  A graph of the solution  .

Remark.  The condition    is not satisfied by the "solution" .  Recall that all solutions involving the use of the Laplace transform are to be considered zero for values of  ,  hence the graph of   is given above in Figure 12.31.  Note that    has a jump discontinuity of magnitude    at the origin.  This discontinuity occurs because either or   must have a jump discontinuity at the origin whenever the Dirac delta function, , occurs as part of the input or driving function.

Explore Solution 12.32.

Taking transforms results in the following equation.

From a table of Laplace transforms we get  , and the solution is  .  

Aside. We can check this with Mathematica's  result using the LaplaceTransform package.

Aside. We use Mathematica/s DSolve package to solve the D.E. and plot the solution.
Notice.  The initial condition for Mathematica must be changed to .  This is required because  .  

(c) 2006 John H. Mathews, Russell W. Howell