Question about multiple functions for a first order ODE

[ArenasField]

The question is as follows:

Suppose you find an implicit solution y(t) to a first order ODE by finding a function H(y, t) such that H(y(t), t) = 0 for all t in the domain. Suppose your friend tries to solve the same ODE and comes up with a different function F(y, t) such that F(y(t), t) = 0 for all t in the domain. Could you both be right, or must one of you be wrong?

I'm pretty lost on this one, honestly. I thought about using the uniqueness theorem (about f(y,t)) being cts and the partial of f(y,t) w.r.t. y), but then I wasn't sure how I could relate this. Can anyone help me out?

 

[BvU]

$y' = 1$ is a first order ODE. $H = y - (t + 4) = 0$ and $F = y - (t + 10) = 0$ both satisfy the ODE, but are different functions. That what you mean ?