Syllabus
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Course Code: MMATH20- 103 Course Name: Ordinary Differential Equations |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Existence and Uniqueness of Solutions: Existence of solutions; Initial value problem, ε-approximate solution, Equicontinuous set of functions, Ascoli lemma, Cauchy–Peano existence theorem and its corollary Uniqueness of solutions; Lipschitz condition, Gronwall’s inequality, Inequality involving approximate solutions, Method of successive approximations, Picard-Lindelöf theorem. Continuation of solutions, Maximal interval of existence, Extension theorem. |
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| 2 | System of linear differential equations: Preliminary definitions and notations. Linear homogeneous
systems; Definition, Existence and uniqueness theorem, Fundamental matrix, Liouville formula, Adjoint
systems, Reduction of the order of a homogeneous system. Non-homogeneous linear systems; Variation of constants formula. Linear systems with constant coefficients. Linear systems with periodic coefficients, Floquet theory. (Relevant portions from the book ‘Theory of Ordinary Differential Equations’ by Coddington and Levinson) |
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| 3 | Theory of linear differential equations: Linear Differential Equation (LDE) of order n, Basic theory of
homogeneous linear equation, Wronskian theory: Definition, necessary and sufficient condition for linear
dependence and linear independence of solutions of homogeneous LDE, Abel’s Identity, Fundamental
sets, More Wronskian theory, Reduction of order. Non-homogeneous linear differential equation of order n: Variation of parameters. Adjoint equations, Lagrange’s Identity, Green’s formula, Self adjoint equation of second order. Linear differential equation of order n with constant coefficients; Characteristic roots, Fundamental set. (Relevant portions from the books ‘Theory of Ordinary Differential Equations’ by Coddington and Levinson and the book ‘Differential Equations’ by S.L. Ross) |
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| 4 | System of differential equations; Preliminary concepts, Differential equation of order n and its equivalent
system of differential equations, Existence and uniqueness of solutions of system of differential
equations. Dependence of solutions on initial conditions and parameters: Preliminaries, continuity and differentiability of solution of a system of differential equations as a function of initial parameters. (Relevant portions from the book ‘Theory of Ordinary Differential Equations’ by Coddington and Levinson) Extremal solutions: Maximal and Minimal solutions. Upper and Lower solutions, Comparison theorems, Existence via upper and lower solutions. Bihari’s inequality. (Relevant portions from the book ‘Textbook of Ordinary Differential Equations’ by Deo et al.) |
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