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Continuous transformations of differential equations with delays. (English) Zbl 0817.34036

The paper deals with transformations \((t,z)= (f_ 1(x, y),f_ 2(x,y))\) of the linear ODE with delays \[ y^{(n)}(x)+ \sum^{n-1}_{i= 0} p_ i(x) y^{(i)}(x)+ \sum^{n-1}_{i= 0} \sum^ m_{j= 1} q_{ij}(x) y^{(i)}(\tau_ j(x))= 0\tag{1} \] to \[ z^{(n)}(t)+ \sum^{n- 1}_{i= 0} r_ i(t) z^{(i)}(t)+ \sum^{n- 1}_{i= 0} \sum^ m_{j= 1} s_{ij}(t) z^{(i)}(\mu_ j(t))=0. \] Under suitable assumptions such a transformation \(z= g(t) y(h(t))\),\(h= f^{-1}_ 1\), \(g= f_ 2(h(t),y(h(t)))\) is constructed. An example of a first-order equation is considered.

MSC:

34K05 General theory of functional-differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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References:

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