Results 171 to 180 of about 149,442 (210)
Linear Differential Equations [PDF]
, 2014 For linear differential equations, the structures of solution spaces are characterized and solution formulas are derived in terms of fundamental matrix solutions. The calculations of fundamental matrix solutions are studied for the constant coefficient case and the Floquet theory is presented for the periodic coefficient case.
Eugene P. Ryan, Hartmut Logemann
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2009
Linear differential equations are simpler in many respects. The truth of this statement is already obvious from the fact that their solution spaces possess the structure of a vector space. Thus it is not surprising that some of our previous results may be improved in this special case.
Richard Bronson, Gabriel B. Costa
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Linear differential equations are simpler in many respects. The truth of this statement is already obvious from the fact that their solution spaces possess the structure of a vector space. Thus it is not surprising that some of our previous results may be improved in this special case.
Richard Bronson, Gabriel B. Costa
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2016
This chapter is devoted to linear differential equations . That is, it presents the solution of first order differential equations using both HF domain differentiation operational matrices and integration operational matrices. Higher order differential equations are also solved via the same first order operational matrices, and again employing one-shot
Gautam Sarkar +2 more
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This chapter is devoted to linear differential equations . That is, it presents the solution of first order differential equations using both HF domain differentiation operational matrices and integration operational matrices. Higher order differential equations are also solved via the same first order operational matrices, and again employing one-shot
Gautam Sarkar +2 more
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2017
In this chapter we obtain corresponding results to those in Chapter 3 for continuous time. More precisely, we study in detail the class of Lyapunov exponents defined by the solutions of a nonautonomous linear equation. In particular, we obtain lower and upper bounds for the Grobman coefficient.
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In this chapter we obtain corresponding results to those in Chapter 3 for continuous time. More precisely, we study in detail the class of Lyapunov exponents defined by the solutions of a nonautonomous linear equation. In particular, we obtain lower and upper bounds for the Grobman coefficient.
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1966
Publisher Summary This chapter focuses on higher-order linear equations. Even for second-order linear equations, no general method of solution is available as there was for first-order equations. Formulas for general solutions can be found for certain special classes of higher-order equations.
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Publisher Summary This chapter focuses on higher-order linear equations. Even for second-order linear equations, no general method of solution is available as there was for first-order equations. Formulas for general solutions can be found for certain special classes of higher-order equations.
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Linear Differential Equations [PDF]
, 1999 In this chapter, we shall first explain the existence of solutions of initial value problems for differential equations and then fundamental theorems for linear differential equations in the complex domain. We explain the definition of regular and irregular singularities of linear differential equations, and the behavior of of local solutions near ...
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1993
Consider the linear differential equation $$ u^{(n)} = p(t)u $$ (1.1) where. As a rule we assume that either $$ p(t) \leqslant 0 for t \in R_ + $$ (1.2) or $$ p(t) \geqslant 0 for t \in R_ + . $$ (1.3) A solution of (1.1) is said to be oscillatory if it has infinitely many zeros.
Ivan Kiguradze, T. A. Chanturia
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Consider the linear differential equation $$ u^{(n)} = p(t)u $$ (1.1) where. As a rule we assume that either $$ p(t) \leqslant 0 for t \in R_ + $$ (1.2) or $$ p(t) \geqslant 0 for t \in R_ + . $$ (1.3) A solution of (1.1) is said to be oscillatory if it has infinitely many zeros.
Ivan Kiguradze, T. A. Chanturia
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Linear Partial Differential Equations
1957Although we shall mainly be concerned in this Part with differential equations, the methods we use here for their discussion and solution are intimately connected with the geometry of the rest of the volume. In particular, the results obtained depend to a great extent on the theory of modules and the intersections of a set of algebraic varieties ...
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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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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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