How to find the IVP of this equation? 
\begin{cases} \dfrac{\mathrm{d}^2y}{\mathrm{d}t^2} + 4y = 2t \\ y(0) = 1 \\ y'(0) = 2 \end{cases}

This is incorrect, but I'm unsure how to find the homogeneous solution $Y_h(t)$. I tried to do:
$Y_h(t) = 2t(A + B)$
$Y_p(t)$ means particular solution:
$Y_p(t) = 2t^2(A + B)$
Then, I do the first derivative and second derivatives:
$Y'_p(t) = 4t(A + B)$
$Y''_p(t) = 4(A + B)$
Once I plug back the derivatives into the original equation, I got:
$4(A + B) + 8t^2(A + B) = 2t$
Could someone please kindly explain what I did incorrectly?
 A: Given that the OP is using $y(p)$ to represent the particular solution, I assume that $y(h)$ represents the solution of the homogeneous equation-- sometimes (in my experience) denoted $y_h(t)$. By definition, the homogeneous solution must satisfy
$$ \frac{d^2}{dt^2}y_h(t)+4y_h(t) = 0.$$ 
This is a linear, second order differential equation, so we assume that $y_h(t) = \exp(\lambda t)$ to arrive at the characteristic equation
$$ (\lambda^2 +4)\exp(\lambda t) = 0, $$
which can hold only if $ \lambda^2 +4 =0$. Thus, we arrive at $\lambda = \pm 2i$, and 
$$ y_h(t) = c_1 \cos(2t) + c_2\sin(2t)$$ 
with unknown coefficients $c_{1,2}$ to be determined from the initial conditions. 
Now, to determine the non-homogeneous solution, $y_p(t)$ it appears as though OP is looking to use the undetermined coefficients method. The non-homogeneous equation is 
$$ \frac{d^2}{dt^2}y_p(t)+4y_p(t) = 2t.$$ 
It is not so hard to see that the right hand side of the non-homogeneous equation is a first degree polynomial. Thus, we guess that the particular solution will also be first degree polynomial and write
$$ y_p(t) = At+b $$.
Inserting this guess into the non-homogeneous equation, and noticing that the second derivative kills the first degree polynomial, gives
$$ 4y_p(t) = 4(At+B) = 2t$$
for $A,B$ unknown constants. 
Collecting common terms in the above equation yields the non-homogeneous solution. Finally, evaluation of $y(t) = y_p(t)+y_h(t)$ at the initial conditions ($t=0$) gives the values of the unknown constants $c_{1,2}$.
