Q1: Consider the ordinary differential equation $\dot{x} = 3 x + 1$. Does a solution for the initial value problem with $x(0) = x_0$ always exist?

A1: Yes!

A2: No, never!

A3: Only for $x_0 \neq 0$.

Q2: Consider the ordinary differential equation $\dot{x} = v(x)$ with continuous $v$. Can it happen that you have two distinct solutions for the initial value problem with $x(0) = 0$.

A1: Yes, for some continuous functions $v$ this can happen.

A2: No, never!

A3: This can only happen for non-continuous functions $v$.

Q3: Consider the ordinary differential equation $\dot{x} = 3 x + 1$. Can it happen that you have two distinct solutions for the initial value problem with $x(0) = 0$.

A1: Yes, one can have more solutions.

A2: No, this cannot happen for this example.

A3: One needs more information.