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Differential Equations - MTH401 - VU Video Lectures

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Course overview.
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Definition of a differential equation, Classification of differential equations, Solution of a differential equation, Existence and uniqueness of solutions.
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Separable equations.
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Homogeneous differential equations.
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Exact differential equations.
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Integrating factor technique.
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First order linear equations.
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Bernoulli equations.
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Applications of first order differential equations : Orthogonal trajectories, Population dynamics.
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Radioactive decay, Newton's law of cooling, Carbon dating.
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Applications of Non linear equations.
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Higher order linear differential equations, Linear differential equation, Non-homogeneous differential equation, Associated homogeneous differential equation.
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Solutions of higher order linear equations.
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Construction of a Second Solution.
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Homogeneous linear equations with constant coefficients.
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Method of undetermined coefficients-superposition approach.
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Undetermined coefficient: Annihilator operator approach.
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Method of undetermined coefficients, Examples.
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Variation of parameters: First order equation, Second order equation.
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Variation of parameters method for higher-order equations.
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Applications of second order differential equation.
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Forced motion – Examples.
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Differential equations with variable coefficients, Cauchy- Euler equation.
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Cauchy-Euler equation: Alternative method of solution.
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Power series, Convergence and Divergence, Ratio test, Absolute convergence, Arithmetic of power series.
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Power series solution of a differential equation.
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Analytic function, Ordinary and singular point, Polynomial coefficients, Non-polynomial coefficients.
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Solution about singular points, Regular and Irregular singular points, Frobenius’ theorem, Method of Frobenius-Cases 1: Roots not differing by an integer.
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Method of Frobenius-Cases II and III: Roots differing by a positive integer and Equal indicial roots.
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Bessel’s differential equation, Series solution of bessel’s differential equation, Bessel’s function of the first Kind.
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Legendre’s differential equation, Legendre’s polynomials, Rodrigues formula for legendre’s polynomials, Legendre’s polynomials are orthogonal, Normality condition for legendre’ polynomials.
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Systems of linear differential equations, Simultaneous differential equations, Systematic elimination: Operator method.
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Solution of using determinants.
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Systems of linear first-order equation, The Nth-Order System, Systems reduced to normal form, Degenerate systems.
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Introduction to Matrices: Matrix, Order of matrix, Square matrix, Equality of matrix, Column matrix, Multiple of matrices, Addition, Difference, Multiplication of Matrices, Zero matrix, Associative, Distributive Laws, Determinant of a matrix.
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The Eigenvalue problem, Eigenvalues and Eigenvectors.
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Matrices and systems of linear first-order equations, Initial–value problem.
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Matrices and systems of linear first-Order equations (continued), Theorem: Existence of a fundamental set, Non-homogeneous systems, Particular integral.
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Homogeneous linear systems, Eigenvalues and Eigenvectors.
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Real and repeated eigenvalues, Eigenvalue of multiplicity m,Two and Three.
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Non-homogeneous system.
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Differential Equations