Please tell me your thoughts about this, and if you agree or disagree. I'll describe my current viewpoint, which is subject to change. Note that I've never taught a lower-division ODEs course.
It seems to me that awareness of a uniqueness theorem for initial value problems is key to understanding the material usually presented in a lower-division ODEs class.
For example, consider the problem of finding all solutions to the differential equation $y'(t) = \lambda y(t)$ (where $\lambda \in \mathbb R$).
This equation is separable and we can solve it using the standard technique for separable differential equations:
\begin{align} & y'(t) = \lambda y(t) \\ \implies & \frac{y'(t)}{y(t)} = \lambda \quad \text{(!)} \\ \implies & \log | y(t) | = \lambda t + C \\ \implies & |y(t)| = e^C e^{\lambda t} \\ \implies & y(t) = e^C e^{\lambda t} \text{ or } y(t) = -e^C e^{\lambda t}. \end{align}
Step (!) is questionable because we might be dividing by $0$. In fact, this derivation did miss the solution $y(t) = 0$. Could there be other solutions we missed, which are equal to $0$ for some but not all values of $t$?
One might worry about other steps as well.
However, the uniqueness theorem for IVPs saves us because we can check that $y(t) = A e^{\lambda t}$ is indeed a solution to our differential equation for any $A \in \mathbb R$, and that any initial condition $y(t_0) = y_0$ could be satisfied with an appropriate choice of $A$. Therefore, we have in fact found all solutions to our differential equation.
The uniqueness theorem for IVPs allows us to be somewhat cavalier in our process of finding solutions. We can make guesses, we can use questionable techniques, but as long as we find a solution "by hook or by crook" that satisfies a given initial condition, the uniqueness theorem tells us that we have found all solutions to the IVP.
I think that if students don't understand the role the uniqueness theorem plays in solving problems like this, they don't really understand the material. Therefore, I think a uniqueness theorem for IVPs should be stated early in a lower division ODEs class.
Do you agree? Is this standard?
Thanks!