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Let $y_1$ and $y_2$ be 2 different continuously differentiable functions in $[a,b]$, both of them solutions to the initial value problem

$$y'=F(x,y),$$ $$y(x_0)=y_0,$$ where $F$ is continuous. Could these be the only 2 solutions of the IVP? If not, what is the minimum number of solutions of the above IVP?

I was looking for specific functions that could satisfy the demands but I feel like it wouldn't help me thoroughly prove prove what is needed. Any help? Thanks.

*It is a copy of a question I asked, now it is with $\LaTeX$.

Yiorgos S. Smyrlis
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Dean
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1 Answers1

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If an IVP possesses two solutions, then it possesses infinitely many solutions. In fact, it possesses a continuity of solutions!

See If an IVP does not enjoy uniqueness, then there are infinitely many solutions.

Yiorgos S. Smyrlis
  • 86,581