Syllabus
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Course Code: MMATH20 -203 Course Name: Differential Equations |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Linear second order equations: Preliminaries, Superposition principle, Riccati’s equation, Prüffer
transformation. Oscillations of second order differential equations: Zero of a solution, Oscillatory and nonoscillatory equations, Abel’s formula, Common zeros of solutions and their linear dependence, Sturm separation theorem, Sturm fundamental comparison theorem and its corollaries, Elementary linear oscillations, Comparison theorem of Hille-Wintner, Oscillations of x// + a(t)x = 0. (Relevant portions from the book ‘Differential Equations’ by S.L. Ross and the book ‘Textbook of Ordinary Differential Equations’ by Deo et al.) |
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| 2 | Second order boundary value problems (BVP): Linear problems; periodic boundary conditions,
regular linear BVP, singular linear BVP; non-linear BVP,
Sturm-Liouville BVP; Definition, Characteristic values and Characteristic functions.
Orthogonality of characteristic functions. Green’s functions: Definition and Properties. Applications of boundary value problems, Picard’s theorem. (Relevant portions from the book ‘Differential Equations’ by S.L. Ross and the book ‘Textbook of Ordinary Differential Equations’ by Deo et al.) |
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| 3 | Non-linear Differential Equations: Autonomous systems; Phase plane, Paths and Critical points,
Types of critical points; Node, Center, Saddle point, Spiral point, Stability of critical points,
Critical points and paths of linear systems; Basic theorems and their applications. Critical points and paths of non-linear systems; Basic theorems and their applications. Non-linear conservative systems, Potential energy function, Dependence on a parameter. Limit Cycles and periodic solutions, Benedixson’s non-existence criterion, Half-path, Limit set, Statement of Poincaré-Benedixson theorem and its uses. (Relevant portions from the book ‘Differential Equations’ by S.L. Ross) |
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| 4 | Stability of linear and non-linear systems: System of equations with constant coefficients, linear
equation with constant coefficients. Lyapunov Stability: Stability of solution of a differential system, Positive definite and semidefinite functions, Negative definite and semidefinite functions, Decrescent function, Lyapunov function, Lyapunov’s theorems on stability. Stability of quasi-linear systems. Boundedness of solutions of a second order differential equations. (Relevant portions from the book ‘Textbook of Ordinary Differential Equations’ by Deo et al). |
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