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Consider the first order differential equation:

$$\begin{equation} \frac{dy}{dt} = \frac{t}{y}, y(0) = 0 \end{equation}$$

Which of the following statements is true?

There exists a unique solution passing through the point $(t=0,y=0)$

There exists a unique solution passing through the point $(t=0,y=0)$ but it may not exist for later times.

There is no guarantee of a unique solution passing through $(t=0,y=0)$

There do not exist any solutions passing through $(t=0,y=0)$