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Note on canonical forms for functional-differential equations. (English) Zbl 0955.34059
The scalar equation \[ y'(x)=p(x)y(x)+q(x)y(h(x)), \qquad x\geq x_0,\tag{1} \] is considered. The author deals with transformations \(t=\varphi(x)\) which change (1) into equations with constant delays.
Using the Abel equation one can transform (1) into (2) \[ v(t)z'(t)=(p(h(t))-w(t))z(t)-p(h(t))z(t-1) \tag{2} \] and \(v,w=O(\lambda^t)\) as \(t\rightarrow\infty\). This result is applied to asymptotic formula for solutions to (1).

MSC:
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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