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

Laitochová, Jitka

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 34, 127-136 (1995).

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.

Reviewer: S.Staněk (Olomouc)

MSC:

34C20	Transformation and reduction of ordinary differential equations and systems, normal forms
34A30	Linear ordinary differential equations and systems

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