Sturm–Picone comparison theorem

In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain. Let pi, qi for i = 1, 2 be real-valued continuous functions on the interval [a, b] and let ( p 1 ( x ) y ′ ) ′ + q 1 ( x ) y = 0 {\displaystyle (p_{1}(x)y^{\prime })^{\prime }+q_{1}(x)y=0} ( p 2 ( x ) y ′ ) ′ + q 2 ( x ) y = 0 {\displaystyle (p_{2}(x)y^{\prime })^{\prime }+q_{2}(x)y=0} be two homogeneous linear second order differential equations in self-adjoint form with 0 < p 2 ( x ) ≤ p 1 ( x ) {\displaystyle 0<p_{2}(x)\leq p_{1}(x)} and q 1 ( x ) ≤ q 2 ( x ) .

Source: Wikipedia — Sturm–Picone comparison theorem (CC BY-SA 4.0)

Sturm–Picone comparison theorem

In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain. Let pi, qi for i = 1, 2 be real-valued continuous functions on the interval [a, b] and let ( p 1 ( x ) y ′ ) ′ + q 1 ( x ) y = 0 {\displaystyle (p_{1}(x)y^{\prime })^{\prime }+q_{1}(x)y=0} ( p 2 ( x ) y ′ ) ′ + q 2 ( x ) y = 0 {\displaystyle (p_{2}(x)y^{\prime })^{\prime }+q_{2}(x)y=0} be two homogeneous linear second order differential equations in self-adjoint form with 0 < p 2 ( x ) ≤ p 1 ( x ) {\displaystyle 0<p_{2}(x)\leq p_{1}(x)} and q 1 ( x ) ≤ q 2 ( x ) .

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Source: Wikipedia "Sturm–Picone comparison theorem" · CC BY-SA 4.0

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