There are no derivatives, therefore this is not a ODE. Secondly (t^2,-t) is not a solution.
Missing derivatives, and the second line should be,
x_2' = x_2/t^2 + 2 x_2/t
What your coarse probably wants from you is to use the following,
\delta(t - a) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i \omega (t -a) } d\omega.
If you want a full but rather simple proof, for a graduate student with a mathematical focus, of this identity I can send it to you.