Differential Equations/Uniquness

Question

Use uniqueness theorem to show that the solution of the initial value problem dy/dt=tsin(y), y(0)=2^(1/2) is positive on all maximal intervals of existence. I'm really lost and don't know how to get started with this problem.

Answer 1

Hint: ignoring the initial condition $y(0) = \sqrt{2}$, the function $\tilde{y}(t) \equiv 0$ is a solution to your differential equation.

Hint 2: argue by contradiction: if your solution is not always positive, then there must be some time $t_0$ such that $y(t_0) = 0$. Use Hint 1 and the uniqueness theorem.