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Decomposing a 4th order linear differential equation as a symmetric product

Mark van Hoeij (2002)

Banach Center Publications

Let L(y) = 0 be a linear differential equation with rational functions as coefficients. To solve L(y) = 0 it is very helpful if the problem could be reduced to solving linear differential equations of lower order. One way is to compute a factorization of L, if L is reducible. Another way is to see if an operator L of order greater than 2 is a symmetric power of a second order operator. Maple contains implementations for both of these. The next step would be to see if L is a symmetric product of...

Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

We give a meaning to derivative of a function u ℝ → X , where X is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space 𝒯 x X of x ∈ X . Let u , v [ 0 , 1 ) → X , u ( 0 ) = v ( 0 ) be continuous at zero. Then by the definition u and v are in the same equivalence class if they are tangent at zero, that is if lim h → 0 + d ( u ( h ) , v ( h ) ) h = 0 . By 𝒯 x X we denote...

Differential equations on the plane with given solutions.

R. Ramírez, N. Sadovskaia (1996)

Collectanea Mathematica

The aim of this paper is to construct the analytic vector fields with given as trajectories or solutions. In particular we construct the polynomial vector field from given conics (ellipses, hyperbola, parabola, straight lines) and determine the differential equations from a finite number of solutions.

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