# ACM10100 Differential & Difference Equations: Applications to Econ & Fin

Differential and difference equation models describe a wide range of complex problems in economics and finance. This course focuses on ordinary differential equations (ODEs) and first order difference equations to develop skills in the formulation, solution, understanding and interpretation of applied problems in finance and economics. A key aim of the course is to develop practical skills that can be applied in a wide range of applied settings. Topics covered are: first order ordinary differential equations, second order ordinary differential equations, first order difference equations, first order systems of two differential equations.

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Curricular information is subject to change

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.

Student Effort Hours:
Student Effort Type Hours
Lectures

36

Tutorial

12

Specified Learning Activities

36

Autonomous Student Learning

36

Total

120

Approaches to Teaching and Learning:
Lectures (36 hours ), Tutorials (12 hours) , Specified learning activities (36 hours), Autonomous student learning ( 36 hours)
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Incompatibles:
ACM10060 - Appl of Differential Equations, MATH20320 - Quantitative Methods Business, MST30040 - Differential Equations

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Exam (In-person): 2 Hour Final Exam n/a Standard conversion grade scale 40% No

70

Assignment(Including Essay): Ten Exercise Sheet Problems to be Submitted Weekly During Tutorials n/a Standard conversion grade scale 40% No

10

Exam (In-person): 1 Hour In-Class Midterm Exam n/a Standard conversion grade scale 40% No

20

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour