?Sum

DSolve[{ y''[t] + y[t]  ==  
          Sum[ DiracDelta[ t - k Pi],  {k,  1,  20}],  
        y[0] == 0,  y'[0] == 0}, 
y,  
      t]

the above output from DSolve was my Out[4]

note: the output is with DSolve[eqns, y, t] and *not* with DSolve[eqns, y[t], t]

?Plot

myy[t_]  =  y[t]  /.  %4[[1]]

Plot[ myy[t],  {t,  -Pi,  6 Pi}]

y[4]  /.  %4[[1]]

Limit[Exp[-t],  t -> Infinity]

Exp[-t]  /.  t -> Infinity

You can use Mathematica to evaluate convolution integrals for you.

Integrate[ 
      Cos[ alpha (t - tau)] Exp[-tau] (1 - 
            Exp[-tau]), 
{tau,  0,  t}]

Note that Mathematica knows that sines, cosines, and exponentials are all faces of the same function.  It uses a shortcut to perform the above integration by looking at it as a path integral in the complex plane rather than an ordinary integral along the real line.  We know the answer is a real number, though, and "Simplify"ing the answer shows it.

Simplify[%]

For greek letters type:

\(α\  = \ \(\(\\)\(\ \)\([\ 
          Alpha\ ]\)\)\)
\(τ\  = \ \(\(\\)\(\ \)\([\ Tau\ ]\)\)\)
\(Γ\  = \ \(\(\\)\(\ \)\([\ 
          CapitalGamma\ ]\)\)\)

Be sure to take out the spaces from the above (I put them there deliberately so that Mathematica wouln't reinterpret them).