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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.

MSC:

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

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