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On a coincidence of the differential equation \(y''-q(t)y=r(t)\) with its associated equation. (English) Zbl 0708.34012

An associate equation to the nonhomogeneous linear differential equation \((*)\quad y''-q(t)y=r(t),\) \(q\in C^ 2(a,b)\), (a,b)\(\subset {\mathbb{R}}\), \(q(t)<0\) for \(t\in (a,b)\), \(r\in C^ 1(a,b)\) of the basis (\(\alpha\),\(\beta\)), \(\alpha^ 2+\beta^ 2>0\) is the equation \((**)\quad y''-Q(t)y=R(t)\) where Q resp. R are defined as \[ Q=q+\frac{\alpha \beta q'}{\alpha^ 2-\beta^ 2q}+\sqrt{\alpha^ 2-\beta^ 2q}(\frac{1}{\sqrt{\alpha^ 2-\beta^ 2q}})'',\quad R=\frac{\alpha r+\beta r'}{\sqrt{\alpha^ 2-\beta^ 2q}}+2\beta (\frac{1}{\sqrt{\alpha^ 2-\beta^ 2q}})'r, \] for \(t\in (a,b)\). The author’s main results are: (1) There exists a one-to-one mapping between the set of all solutions y of (*) and the set of all solutions Y of (**) given by the relation \(Y=(\alpha y+\beta y')/\sqrt{\alpha^ 2-\beta^ 2y}\); (2) The differential equation (*) coincides with its associate equation (**) by the basis (\(\alpha\),\(\beta\)) exactly if the functions q and r assume in (a,b) some special expressions (for details look at the note).
Reviewer: H.Ade

MSC:

34A30 Linear ordinary differential equations and systems
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References:

[1] Borůvka O.: Linear Differential Transformations of the Second Order. The English Univ. Press, London 1971. · Zbl 0218.34005
[2] Laitoch M.: L’équation associée dans la théorie des transformations des équations différentielles du second ordre. Acta Univ. Palackianae Olomucensis, F.R.N. 12 (1963), 45-62. · Zbl 0256.34005
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