Example 2

In[50]:=

  Clear[de,tde,newtde,Y,y]

Here's the differential equation:

In[51]:=

  de=y''[t]+4 y[t]==t^2

Out[51]=

                      2
  4 y[t] + y''[t] == t

Here is the transformed differential equation:

In[52]:=

  tde=LaplaceTransform[de,t,s]

Out[52]=

                                    2
  4 LaplaceTransform[y[t], t, s] + s  LaplaceTransform[y[t], t, s] - 
   
                       2
     s y[0] - y'[0] == --
                        3
                       s

Let's make use of the initial values:

In[53]:=

  newtde=tde/.{y[0]->0,y'[0]->0}

Out[53]=

                                    2
  4 LaplaceTransform[y[t], t, s] + s  LaplaceTransform[y[t], t, s] == 
   
    2
    --
     3
    s

Now solve this equation for Y[s], which Mathematica calls
LaplaceTransform[y[t],t,s]. The extra commands are to make the solution be a function, not a list of rules.

In[54]:=

  Solve[newtde,LaplaceTransform[y[t],t,s]];
  Y[s_]:=Evaluate[LaplaceTransform[y[t],t,s]/.Flatten[%]]
  Y[s]

Out[54]=

       2
  -----------
   3       2
  s  (4 + s )

Now transform back to find y(t).

In[55]:=

  y[t_]:=InverseLaplaceTransform[Y[s],s,t]
  y[t]

Out[55]=

          2
    1    t    Cos[2 t]
  -(-) + -- + --------
    8    4       8

Check:

In[56]:=

  Simplify[D[y[t],{t,2}]+4 y[t]]

Out[56]=

   2
  t

Up to Solving Constant Coefficient Differential Equations (Initial Value Problems)