does a linear differential equation have a well defined initial value problem if a term diverges at initial 'time'? Suppose I have a differential equation like the following: 
$$\frac{d^2x}{dt^2}+t^2x=0$$
And I've to put initial conditions at $t=-\infty$. Now the $t^2$ bit seems to diverge at $t=-\infty$. Is that a problem? More generally can I put initial conditions at any time t if one or more of the coefficients of the equation diverges at t?
My feeling is that we need to find the solutions (parabolic cylindrical functions in this case apparently) and if they are well behaved at the initial time then there is no problem. Is that correct?
Note: I am a physicist, so please keep your answers a bit simple :)
Edit: In this paper one finds an example of such an equation being applied http://arxiv.org/pdf/0806.2496v1.pdf (equation 59-62).  

$$\frac{d^2x}{dt^2}+t^2x=0$$
Changing $t$ to $-t$ doesn't change $x(t)$. Hense $x(t)$ is an even function.
The general solution can be expressed in terms of particular parabolic cylinder functions. But the variable of those functions are on the complex range, which leads to more complicated interpretation.
The ODE is also a generalized form of Bessel ODE. The general solution can be expressed in terms of particular Bessel functions (related to the above parabolic cylinder functions. But doesn't matter, the Bessel form of solution can be derived directly) :
$$x(t)=c_1\sqrt{|t|}J_{1/4}\left(\frac{t^2}{2}\right)+c_2\sqrt{|t|}J_{-1/4}\left(\frac{t^2}{2}\right)$$
The function $x(t)$ is real and finite for all real $x$, insofar the coefficients $c_1$ and $c_2$ are real.
$t\to \pm\infty \quad x(t)\to 0$ , asmptotically: 
$\sqrt{|t|}J_{\pm 1/4}\left(\frac{t^2}{2}\right) \sim \frac{2}{\sqrt{\pi}} \sin\left(\frac{t^2}{2}+\frac{\pi}{8} \right)\frac{1}{\sqrt{|t|}}$
$t=0 \quad \begin{cases}
\sqrt{|t|}J_{-1/4}\left(\frac{t^2}{2}\right) = \frac{\sqrt{2}}{\Gamma(\frac{3}{4})} \quad \text{; derivative}=0\\
\sqrt{|t|}J_{1/4}\left(\frac{t^2}{2}\right) =0 \quad \text{; discontinuous derivative}
\end{cases}$
