Question: If $f$ and $g$ have compact support, then does it follow that $f \* g$ has compact support?

Answer: 
First realize that $(t^2)''=2$, hence let:
$$4x(t)=4y(t)-t^2+\frac{1}{2}$$

$$\Rightarrow x''(t)=y''(t)-\frac{1}{2}$$

The equation becomes:

$$x''(t)=-4x(t)$$

Note that this can be seen as a differential equations relating the position($x(t)$) and acceleration

($a(t)=x''(t)$) of an object in harmonic motion, the solution of which:
$$x(t)=Acos(2t + \varphi)$$
with constants $A,\varphi $

substitue back to get the answer
The forcing function on the right side is of degree two, and you'll need to add two extra degrees - one for each derivative in the highest occurrence - to do this right. That is, you need to assume $y = ax^4 + bx^3 + cx^2 + dx + e$ to start.