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Is there literature available on solving differential equations of the type $$f(x,y(x),y(\kappa x),y'(x))=0,$$ where $\kappa$ is a given constant? I know about the book Introduction to the Theory and Application of Differential Equations with Deviating Arguments by L.E. El'sgol'ts and S.B. Norkin from the year 1973 [1], but I wonder if there are more recently published books available as well.

Specifically, I would be interested in solving for example $$u(2t)-2u'(t)u(t)=0$$ without guesswork.

[1] Introduction to the Theory and Application of Differential Equations with Deviating Arguments, L.E. El'sgol'ts and S.B. Norkin, Mathematics in Science and Engineering, Volume 105, Academic Press, New York, 1973

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  • $\begingroup$ See my question math.stackexchange.com/q/2071653/389818 I didn't deem the answer to be too helpful. $\endgroup$
    – Teh Rod
    Jan 3, 2017 at 19:36
  • $\begingroup$ By the way, $u=t$ is a solution as well in this case. So the uniqueness conditions are probably different for such equation $\endgroup$
    – Yuriy S
    Jan 18, 2018 at 12:44

1 Answer 1

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In regards to more recent literature (more recent than this question in fact), one interesting paper may be "A new kind of functional differential equations" by Kong and Zhang. This covers your situation, given $\kappa \in (0,1)$, as well as some more general functional delays. The approach in this paper is that of (the unfortunately named) Retarded Delay Differential Equations, where the delayed functions are not derivatives. To cover the case of $\kappa \in (1,\infty)$, you may follow a similar approach with Advanced Delay Differential Equations, where the delayed functions may include derivatives, by doing a change of variables $x\rightarrow \xi/\kappa$. You may need some background in geometric flow theory for Section 1.3, but outside of that it doesn't seem to require any differential geometry knowledge. The paper covers both ODEs and PDEs.

In general, this question falls under the blanket of Functional Differential Equations, though Delay Differential Equations is the current poster-child for the subject. Usually those delays come in the form of $\phi (t) = t-\tau$ for some constant $\tau \in (0,t)$. However, more research into general delays is up and coming, and may be a popular area of DE research in the coming years.

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