## Calculus and Applications B

 Unit code: MATH10232 Credit Rating: 15 Unit level: Level 1 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

Co-Requisite

#### Aims

This course unit aims to introduce students to ordinary differential equations, primarily covering methods of solution and applications to physical situations.

#### Overview

The unit will cover first and second order ordinary differential equations including classification and standard solution methods. Applications will be drawn from the field of classical mechanics, but no prior experience in mechanics is expected or required. Matlab will be used to illustrate some of the ideas and methods.

#### Learning outcomes

On completion of this unit successful students will be able to solve first order and second order linear problems and first order separable equations analytically. Use substitution methods and power series methods to find solutions. Be able to investigate solutions using direction fields and Euler's method. Have used Matlab as a mathematical tool and used differential equations to solve problems in mechanics and other applications.

#### Assessment methods

• Other - 25%
• Written exam - 75%

#### Assessment Further Information

Coursework; Weighting within unit 15%

Supervision; Weighting within unit 10%

Two hours and half end of semester examination; Weighting within unit 75%

#### Syllabus

1.Introduction: concept of mathematical modeling; definition and classification of ordinary differential equations; order; linear and autonomous equations.

2.First-order ordinary differential equations: separable, graphical and numerical solutions; initial conditions; direction fields; Euler's method; use of Matlab; existence and uniqueness of solutions; integrating-factor methods; power series; substitution methods for nonlinear equations; phase plane and stability; population modelling.

3.Higher-order ordinary differential equations: general solution of linear, second-order equations; initial and boundary conditions; homogeneous equations; particular integrals; method of undetermined coefficients; harmonic oscillators; resonance; coupled systems; phase portraits; substitution methods for nonlinear equations; plane autonomous systems; predator-prey systems.

4.Mechanics: particle kinematics in Cartesian and polar coordinates; Newton's laws; forces; Newton's law of gravitation; work and energy; motion along a line (potential wells); equilibrium and stability; simple systems of particles; simple pendulum; compound pendulum; double pendulum.

James Stewart, Calculus, Early Transcendentals, Thomson, 5th Edition, International Student Edition, 2003.

C. H Edwards, Elementary differential equations with boundary value problems, Pearson Prentice Hall, 2004.

#### Feedback methods

Feedback supervision will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Joel Daou - Unit coordinator