Solving a non-homogenous IVP The question I am working on is:
Find the solution of:
$y'' + 4y' + 4y = 216e^{4t}$
with $y(0) = 4$ and $y'(0) = 6$
......
I assumed I would have to get it into a general form, along the lines of 
$y(t) = c_1e^{-2t} + c_2te^{-2t}$ 
r = -2, and -2, so I put them in the complementary solution. From there I expect I would take the derivative of y(t) and then use the initial conditions to find c1, and then c2. 
The extra 't' is there because it's a repeated root. It's a basic IVP. 
However, I'm not sure how to approach this because of the 216e^(4t), and that's what's throwing me off. Can someone point me in the right direction and tell me where to go from here? I'm sure I can do this, I think I'm missing a few steps though. 
 A: You can use the method of undetermined coefficients. Assume $Y=Ae^{4t}$ solves the nonhomogeneous equation. Take its derivatives, plug into the ODE, and solve for $A$. Then your general solution is
$$y=c_1 e^{-2t}+c_2 t e^{-2t}+Ae^{4t}.$$
Now you solve for $c_1$ and $c_2$ using the initial conditions.
Note that it's really important that you solve for $A$, the unknown coefficient, before solving the the $c$'s using the initial conditions.
A: You can solve this using the method of undetermined coefficients.
You know a general solution $y_h$ to the homogeneous problem
$$
y'' + 4y' + 4y = 0
$$
To find a solution to your non-homogeneous problem, you need to find a particular solution $y_p$ of your differential equation, and then write
$$
y = y_h + y_p = c_1 e^{-2t} + c_2 t e^{-2t} + y_p
$$
and then use the initial conditions to determine $c_1$ and $c_2$.
The method of undetermined coefficients (you should look up the full details) makes a guess at $y_p$, then uses the ODE to narrow things down. In particular, for this problem, you guess that $y_p = Ae^{4t}$ (based on the form of the left-hand side and the homogeneous solution; again, you should do some reading on the particulars of this process).
Then, you plug $y_p$ into the equation, and determine what $A$ needs to be in order for 
$$
y_p''+ 4y_p' + 4y_p = 216 e^{4t}
$$
Then go back and solve for $c_1,c_2$.
A: Hint:
$$
(e^{2t}y)''=e^{2t}(y''+4y'+4y).
$$
