Given a first-order differential equation $\frac{dy}{dx} = y^4$ with no IVP, I am to argue whether or not there exists a unique solution.
My immediate thought was that of course there is no unique solution. We can solve the differential equation for a general solution by separating the variables. This general solution would then correspond to many particular solutions for an arbitrary choice of integration constant. The particular solutions are obviously distinct for different constants meaning distinct solutions to the differential equation (not unique). Similarly, $y=0$ is also a valid solution.
However, both my professor and the TA argue that by the Picard-Lindelöf theorem the differential equation has a unique solution as both $y^4$ and its partial derivative are continuous on all real numbers. However, I am under the impression they are misinterpreting what is asked as the Picard-Lindelöf theorem states that the solution is unique to a given initial value problem. As we have no initial constraints the theorem does not really apply.
Here are some definitions of “unique” and “solution to a differential equation” from a quick google search:
- Unique: In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition (from Wikipedia)
- Solution: A solution of a differential equation is an expression for the dependent variable in terms of the independent one(s) which satisfies the relation (lecture notes from University of Glasgow)
I would argue that these definitions prove my points, but am I misunderstanding something?