Question about multiple functions for a first order ODE

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SUMMARY

The discussion centers on the uniqueness of solutions to first order ordinary differential equations (ODEs). Specifically, the participants analyze the functions H(y, t) = y - (t + 4) and F(y, t) = y - (t + 10), both of which satisfy the ODE y' = 1. The uniqueness theorem states that if the function f(y, t) is continuous and its partial derivative with respect to y is continuous, then the solution to the ODE is unique. Therefore, while both functions satisfy the ODE, they cannot both represent the same solution due to the uniqueness condition.

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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?
 
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##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 ?
 

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