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On the Meromorphic Solutions of Linear Differential Equations
Journal of Systems Science and Complexity, 2007zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On liouvillian solutions of linear differential equations
Applicable Algebra in Engineering, Communication and Computing, 1992Let \(L(y)=0\) be a homogeneous linear differential equation and \(R(z)=0\) be the associated Riccati equation. The paper deals with the problem of finding the possible degrees of the minimal polynomial of an algebraic solution of \(R(z)=0\). Supposing that \(L(y)\) is irreducible and \(L(y)=0\) has a Liouvillian solution, the author obtains sharp ...
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On Linear Perturbation of Non-Linear Differential Equations
Canadian Journal of Mathematics, 1954In the theory of the asymptotic solution or stability of ordinary differential equations most attention has been given to linear or nearly-linear cases. Investigations in this field, starting primarily with those of Kneser(7)on the equation y″+f(x)y= 0, have by now mostly been summed up in results on the vector-matrix system dy/dx =Ay +f(y,x ...
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2016
In this chapter we deal with some basic facts concerning ordinary linear differential equations in the analytic domain, culminating in Fuchs’ theory on regular singular points.
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In this chapter we deal with some basic facts concerning ordinary linear differential equations in the analytic domain, culminating in Fuchs’ theory on regular singular points.
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2017
This chapter begins by proving the fundamental theorem on first-order ordinary differential equations in normal form, and by studying the differentiability of the solution. As a next step, it derives the canonical form of linear second-order ordinary differential equations with variable coefficients, which is very useful for qualitative investigations.
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This chapter begins by proving the fundamental theorem on first-order ordinary differential equations in normal form, and by studying the differentiability of the solution. As a next step, it derives the canonical form of linear second-order ordinary differential equations with variable coefficients, which is very useful for qualitative investigations.
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2001
Here we consider a so called (scalar) normal system of n ordinary linear differential equations which is a system of the form $$\left\{ {\begin{array}{*{20}{c}} {{{{x'}}_{1}} = {{a}_{{11}}}(t){{x}_{1}} + {{a}_{{12}}}(t){{x}_{2}} + ... + {{a}_{{1n}}}(t){{x}_{n}} + {{f}_{1}}(t),} \\ {{{{x'}}_{2}} = {{a}_{{21}}}(t){{x}_{1}} + {{a}_{{22}}}(t){{x}_{2}} +
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Here we consider a so called (scalar) normal system of n ordinary linear differential equations which is a system of the form $$\left\{ {\begin{array}{*{20}{c}} {{{{x'}}_{1}} = {{a}_{{11}}}(t){{x}_{1}} + {{a}_{{12}}}(t){{x}_{2}} + ... + {{a}_{{1n}}}(t){{x}_{n}} + {{f}_{1}}(t),} \\ {{{{x'}}_{2}} = {{a}_{{21}}}(t){{x}_{1}} + {{a}_{{22}}}(t){{x}_{2}} +
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Remarks on the Linearization of Differential Equations
IMA Journal of Applied Mathematics, 1977openaire +2 more sources
1967
In this section we consider the vector equation $$ \mathop x\limits^. {\rm{ }} = {\rm{ }}A(t){\rm{ }}. $$ (58.1) We assume that the elements a i k (t) of the matrix A are continuous functions of t defined for t ≥ t0. Equations with a constant A, treated in sec. 4, are special cases of (58.1). On the other hand, (58.1) is a special case of the
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In this section we consider the vector equation $$ \mathop x\limits^. {\rm{ }} = {\rm{ }}A(t){\rm{ }}. $$ (58.1) We assume that the elements a i k (t) of the matrix A are continuous functions of t defined for t ≥ t0. Equations with a constant A, treated in sec. 4, are special cases of (58.1). On the other hand, (58.1) is a special case of the
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