Associated linear differential equations to linear second order differential equations of Sturm, Jacobi and general form. (English) Zbl 0855.34045

The homogeneous linear differential equations (ab) \(y'' + a(t) y' + b(t)u = 0\), (AB) \(Y'' + A(t)Y' + B(t)Y = 0\), are considered, where \(a,b,A \in C^2 (J)\). Let \(\alpha, \beta \in \mathbb{R}\), \(\alpha^2 + \beta^2 > 0\), \(t_0 \in J\). The author presents formulas for \(\rho \in C^2 (J)\) and \(B \in C^0 (J)\) \((\rho = \rho (\alpha, \beta, a,b,A)\), \(B = B (\alpha, \beta, a,b,A))\) that, for any solution \(y\) of (ab), the function \[ Y(t) = \rho (t) \Bigl( \alpha y(t) + \beta e^{\int^t_{t_0} a(s) ds} y'(t) \Bigr),\;t \in J \] is a solution of (AB). The same problem for equations \((p(t)y') - q(t)y = 0\), \((P(t) Y')' - Q(t)Y = 0\) with \(p,q,P \in C^2(J)\), \(p(t) P(t) \neq 0\) on \(J\) is considered as well.


34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A30 Linear ordinary differential equations and systems
Full Text: EuDML


[1] Borůvka O.: Linear Differential Transformations of the Second Order. The English University Press, London, 1971. · Zbl 0218.34005
[2] Laitoch M.: L’équation associée dans la théorie des transformation différentielles du second ordre. Acta Univ. Palacki. Olomuc., Fac. rer. nat. T.12, 1963, 45-62. · Zbl 0256.34005
[3] Laitochová J.: On transformations on two homogenous linear second order differential equations of general and Sturm forms. Acta Univ. Palacki. Olomuc., Fac. rer. nat. 85, Math. 25 (1986), 77-95. · Zbl 0643.34048
[4] Fialka M.: On coincidence of the differential equation y” - q(t)y = r(t) with its associated equation. Acta Univ. Palacki. Olomuc., Fac rer. nat. 91, Math. 27 (1988), 251-262. · Zbl 0708.34012
[5] Staněk S., Vosmanský J.: Transformations of linear second order ordinary differential equations. Arch. Math. (Brno) 22, 1 (1986), 55-60. · Zbl 0644.34029
[6] Votava M.: Některé vlastnosti průvodních rovnic s váhovými funkcemi. Acta Facultas Pedagogicae Ostraviensis 92, Series A-20 (1985), 43-51.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.