## 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).

### MSC:

 34A30 Linear ordinary differential equations and systems

### Keywords:

nonhomogeneous linear differential equation
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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 [3] Laitoch M.: Homogene lineare zu sich selbst begleitende Differentialgleichung zweiter Ordnung. Acta Univ. Palackianae Olomucensis, F.R.N., 33 (1971), 61-72. · Zbl 0298.34006 [4] Staněk S.: Two-point boundary problem in a second order nonhomogeneous linear differential equation. Acta Univ. Palackianae Olomucensis, F.R.N., 61 (1979), 59-71. · Zbl 0437.34012 [5] Stěpanov V. V.: A course of Differential Equations. (in Czech), přírodov. nakl. Praha 1952.
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