Template for solving 2nd order constant coefficient linear differential equations

Part 0. The equation:

Suibstitute your data for the following:

a = 3
b = 5
c = 10
f[t_] := (E^t) (t^2)

It is required to solve the equation

eqn  =  (a y''[t]  +  b y'[t]  +  
            c y[t]  ==  f[t])

Part 1.  The general solution

Execute the following cell to find the general solution to the equation:

Clear[y,  F]
eqn
Soln  =  DSolve[eqn,  y[t],  t];
F[t_]  =  Expand[ Simplify[y[t] /.  Soln[[1]]]]

Part 2.  Initial value problems.

Introduce initial conditions (substitute your data in the following):

Clear[y,  InitialConditions]
InitialConditions  =  {
y[0]  ==  
          3,  
y'[0]  ==  
          2
}

Execute the following cell to solve the initial value problem:

Clear[ F]
Soln  =  
        DSolve[{eqn,  InitialConditions},  y[t],  
          t];
F[t_]  =  Expand[ Simplify[y[t] /.  Soln[[1]]]]