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. |
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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. |
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