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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