I'm starting the course of ordinary differential equations and I'm not really sure if $y'(x) = y(y(x))$ is an ODE. More than the answer what I need is an explanation of the reason of it.
Thanks!
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1The last quote in this post says No. But it gives no reason. – M. Winter Sep 18 '17 at 14:57
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2An ODE of first order can be written as $y'=F(x,y).$ Can you write $y(y(x))=F(y(x))?$ – mfl Sep 18 '17 at 14:57
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1@mfl One should be very precise about the types of objects here, since in some sence I can write $y(y(x)) = F(y(x))$, just take $F=y$ (lexically). – lisyarus Sep 18 '17 at 15:00
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@lisyarus Yes, you are right. I mean $F$ independent of $x,y.$ Thank you for pointing it. – mfl Sep 18 '17 at 15:02
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If $y$ is a constant function, then yes. – G Tony Jacobs Sep 18 '17 at 15:03
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Usually an $n$-th order ODE is defined as a function $F:\Bbb R\times\Bbb R^n\to \Bbb R$ and a pair $(x_0;y_0,y_1,...,y_{n-1})\in\Bbb R\times\Bbb R^n$. The fact that ODEs talk about functions only comes into the game when we define what we consider a solution of this ODE. A solution is an $n$ times differentiable function $y:\Bbb R\to\Bbb R$ with
$$y^{(k)}(x_0)=y_k\text{ for }k=0,...,n-1,\qquad y^{(n)}(x)=F(x;y(x),y'(x),...,y^{(n-1)}(x)).$$
But $y(y(x))$ only makes sense when we already talk about functions when stating the ODE.
Conclusion. So I do not consider $y'(x)=y(y(x))$ an ODE, but I think this is still debatable. It is some "weird" problem statement mixing elements from differential and functional equations.
M. Winter
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Do you mean that the composition doesn't make sense because at first we are consideing y(x) a variable of function F? – Sergi De la Torre Sep 18 '17 at 15:09
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@SergiDelaTorre Yes. Somehow. $y$ in $F$ is just a real number. You cannot compose real numbers. You can state an ODE without even mentioning functions. – M. Winter Sep 18 '17 at 15:10
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Looks like you have a typo: $y$ should be $\mathbb{R}\rightarrow\mathbb{R}$. – lisyarus Sep 18 '17 at 15:21
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1@lisyarus Thank you very much! There were actually more mistakes but I fixed them hopefully! – M. Winter Sep 18 '17 at 15:27
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I think that a possible good term for this would be a functional-differential equation :) – lisyarus Sep 18 '17 at 15:30
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@lisyarus Actually, I think checking whether $y'(x)=y(y(x))$ has any non-trivial solutions besides $y(x)=0$ might be an interesting problem! – M. Winter Sep 18 '17 at 15:33
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