Learning Outcomes:
Upon completion of this module the student should be able to:
identify an ordinary differential equation and classify it by order or linearity.
determine whether or not a unique solution to a first-order initial-value problem exists.
understand differences between solutions of linear and non-linear first-order differential equations.
recognize and solve linear, separable and exact first-order differential equations.
use substitutions to solve various first-order differential equations.
recognize and solve autonomous first-order differential equations, analyze trajectories, and comment on the stability of critical points.
model and solve application problems using linear and non-linear first-order differential equations, including, but not limited to, topics such as: Dynamics of market price, Solow growth model.
determine whether or not a set of solutions to a differential equation are linearly dependent or independent using the Wronskian
use reduction of order to find a second solution from a known solution
solve homogeneous linear equations with constant coefficients
use the method of undetermined coefficients to solve nonhomogeneous linear differential equations for which the nonhomogeneous term can be annihilated
solve nonhomogeneous linear differential equations using variation of parameters
model, solve and analyze problems involving second-order linear differential equations with application t inflation and unemployment models.
find trajectories associated with, determine critical points of, and perform phase plane analyses for simple autonomous linear and non-linear systems of equations.
solve first order difference equations and analyse stability of equilibria.
model and solve application problems using difference equations including market models with inventory
Indicative Module Content:
First Order Ordinary Differential Equations:
Introduction to first order linear ordinary differential equations with constant coefficients.
First order linear differential equations with variable coefficients.
Nonlinear ordinary differential equations of first order.
Solution of first order linear and nonlinear ordinary differential equations by separable variables method.
Exact linear and nonlinear ordinary differential equations, Integrating factor.
Solution of first order linear ordinary differential equations by integrating factor method,
Equation reducible to linear form: Bernoulli’s equation.
Economic applications: Dynamics of market price, Solow growth model.
Second Order Ordinary Differential Equation
Second order linear ordinary differential equations with constant coefficients.
Applications: A market model with price expectations, The interaction of inflation and unemployment.
Second order linear ordinary differential equations with variable coefficients. Variation of constants and undetermined coefficient techniques to construct solutions
First Order Difference Equations:
Discrete time,
Differences and difference equations.
Solving a first order linear difference equation.
Conditions for dynamic stability of equilibrium. The Cobweb model. Market Model with Inventory.
First-Order Systems of two Ordinary Differential Equations
Solving simultaneous dynamical equations.
Phase plane analysis.
Inflation - Unemployment models.