Every ODE(Ordinary Differential Equation) with initial condition has either unique solution or no solution or infinite many solutions. I want an example of an ODE( clearly without initial conditions) with exactly two solutions. Please provide me a simple example . I am unable to find that . Thank you.
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Your first statement about every ODE with initial conditions having either 0,1,$\infty$ solns cannot be correct. e.g. Consider $y'(y'-1)=0,y(1)=1$.Perhaps you were referring to linear ODEs? – Golden_Ratio Jan 06 '22 at 08:24
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@Golden_Ratio ok then ODE given by you has more than one but finitely many solutions? – neelkanth Jan 06 '22 at 08:36
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You can verify its solutions are $y(x)=1,y(x)=x$. – Golden_Ratio Jan 06 '22 at 08:37
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@Golden_Ratio https://math.stackexchange.com/questions/657284/if-an-ivp-does-not-enjoy-uniqueness-then-it-possesses-infinitely-many-solutions – neelkanth Jan 06 '22 at 08:38
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@Golden_Ratio what about This question link ? – neelkanth Jan 06 '22 at 08:44
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https://math.stackexchange.com/questions/1568783/when-does-an-initial-value-problem-have-exactly-two-solutions – neelkanth Jan 06 '22 at 08:56
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many thanks for sharing. But in that first link, in the solution alluding to Hellmuth Kneser, can I ask how does that proof ensure the newly constructed solution is differentiable at $t_0$? Applying that proof to my example would result in sharp points, for instance, if I have understood their construction correctly. – Golden_Ratio Jan 06 '22 at 09:00
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@Golden_Ratio may it’s about ODE of type $y’=(x,y),y(a)=b$ only . – neelkanth Jan 06 '22 at 09:05
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yea I just realized that myself. So can we agree my example suffices as a counterexample to your first statement? – Golden_Ratio Jan 06 '22 at 09:06
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@Golden_Ratio these two solution $x$ and $1$ are clear but how can we say that there no other solutions? – neelkanth Jan 06 '22 at 09:07
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well heuristically, if there are other solutions, they have to be piecewise either of the form $y=$constant or $y=x$+constant, correct? The issue is non-differentiability at sharp points where the pieces meet. – Golden_Ratio Jan 06 '22 at 09:10
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@Golden_Ratio seems to be correct. I will analysis it and comment you ? – neelkanth Jan 06 '22 at 09:12
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Let us continue this discussion in chat. – Golden_Ratio Jan 06 '22 at 09:13