# If #F(x)# and #G(x)# both solve the same initial value problem, then is it true that #F(x) = G(x)#?

Please see below for a discussion of "uniqueness of solutions". and an answer to the question.

The question can be rephrased as:

or

Can there be more that one solution? (This would be non-unique solutions.)

So the phrase Uniqueness of solutions is a heading telling us what the question is about. It is not a part of the question.

So the solution is unique.

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This is asking, "can there be two solutions to the same first-order differential equation?" Every time we formulate a differential equation, we should ask ourselves,

Their arbitrary constants don't necessarily have to be the same, so we suppose they are not for now. They both are given the same initial condition:

And we have

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Yes

We have :

It satisfies the equation:

And using the initial condition

So we have:

And we therefore conclude that

ie the solution to a First Order Differential Equation is Unique

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If ( F(x) ) and ( G(x) ) both solve the same initial value problem, it is not necessarily true that ( F(x) = G(x) ). While both functions may satisfy the initial condition(s) of the problem and produce identical outputs for a particular input, there could still be differences between the functions themselves. These differences could arise due to various reasons such as the methods used for solving the problem, the domain of definition, or the presence of arbitrary constants in the solutions. Therefore, while both functions yield the same results under the specified initial conditions, they may not be identical functions.

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