Introduction to mathematical economics
From Wikiversity
I. Introduction to Differential Equations. First Order Equations.
Introduction. What is a differential equation? Definitions: order, degree, solution. Existence theorem. Operators and operator theory. Variables separable, simple substitutions, homogenous functions, exact differentials, exact differential equations, integrating factors, linear differential equations, linear but not homogeneous equations, other special cases. Applications.
II. Higher Order Differential Equations.
First order equations of degree higher than one. Envelopes. Linear independence. Linear differential equations, homogeneous linear differential equations with constant coefficients, equations in which the auxiliary equation has repeated roots, auxiliary equations have imaginary roots, right member not zero, method of undetermined coefficients, method of variation of parameters. Special cases. Reduction of order by substitution, absent variable cases, Euler’s linear equation, factorization of the operator.
III. Difference Equations: Discrete Time Models.
Discrete time, differences, and sampling. The Difference Calculus. Interpolation and extrapolation. Integration and summation. Differentiation and integration of sums, the Gamma function, Bernoulli numbers and polynomials, Euler numbers and polynomials. Difference equations, differential equations as limits of difference equations, first order difference equations, homogeneous equations, operator methods, method of undetermined coefficients, method of variation of parameters, method of reduction of order. Equations with variable coefficients. Dynamic stability of equilibria, the cobweb model. Extensions to stochastic cases.
IV. Advanced Topics. (As time permits)
Systems of Differential Equations, Concepts and theory. Linear homogenous systems with constant coefficients. Phase plane, critical paths, and stability. Series Solutions and Special Equations. Laplace Transforms and Related Topics. Special functions and equations, including the Legendre and Bessel equations. Nonlinear Systems, Discrete Dynamical Systems and Chaos.
