## MAS290 Methods  for  Differential Equations (NJTech)

 Semester 2, 2016/17 10 Credits Lecturer: Prof Neil Strickland Home page Reading List Aims Outcomes Assessment Full Syllabus

Differential equations arise in most models of real phenomena, including particle mechanics, biology and economics. The first half of this module covers first order differential equations in two variables, with emphasis on geometric features of the phase diagram, linearisation near equilibrium points, and stability analysis using Lyapunov functions. The second half of the course covers power series methods for solution of second order linear equations near ordinary points or regular singular points.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Aims

• To consolidate previous knowledge and develop level 1 work.
• To develop analytical techniques for solving systems of first order differential equations.
• To introduce phase portraits of two-dimensional systems.
• To extend the students' knowledge of linear and non-linear ordinary differential equations.

## Learning outcomes

• Sketch and interpret phase portraits for two dimensional ordinary differential equations.
• Find equilibrium points, and classify them by linearisation.
• Use conserved quantities and Lyapunov functions to study stability and other qualitative features.
• Determine whether second order linear equations have ordinary points, regular singular points or irregular singular points.
• Understand how the indicial polynomial controls the features of solutions near a regular singular point.
• Use series methods to solve equations near an ordinary or regular singular point.
• Use reduction of order to find a second solution.
• Convert equations to normal form, Sturm-Liouville form, or other convenient forms.
• Understand and exploit orthogonality properties of Sturm-Liouville solutions.

32 lectures, 32 tutorials

## Assessment

One formal 2 hour written examination. All questions compulsory.

## Full syllabus

1. Planar ordinary differential equations

• Examples of planar systems, with animations and qualitative discussion.
• Linear systems. Eigenvalues. Fundamental solutions using eigenvectors or other formulae.
• Classification by eigenvalues, or by trace and determinant. Qualitative formulae.
• Stability for linear systems.
• The Jacobian matrix, and linearisation near equilibrium points.
• Local topological conjugacy, and the Hartman-Grobman Theorem.
• Conserved quantities.
• Positive and negative (semi-)definite functions. The quadratic case.
• Strong and weak Lyapunov functions. Implications for stability.
2. Second order linear differential equations
• Reminder of the constant coefficient case, including problems with boundary conditions.
• Preview of important examples, including the Bessel equation, the Lagrange equation, the Hermite equation and the Airy equation.
• Qualitative discussion of general features including orthogonality properties and alternation of roots.
• Power series solutions near ordinary points.
• Radius of convergence.
• Regular singular points, and roots of the indicial polynomial.
• Differing behaviour when the gap between roots is zero, or a nonzero integer, or a non-integer.
• Examples of series solutions with regular singular points.
• Reduction of order, to find a second solution after having found the first one.
• Equations in Sturm-Liouville form. The Wronskian. Eigenfunctions.
• Orthogonality and reality properties. Analogy with real symmetric matrices.
• Normal form.
• Various other transformation to convert equations to forms that are already understood.

## Reading list

Type Author(s) Title Library Blackwells Amazon
B Teschl Ordinary Differential Equations and Dynamical Systems Blackwells Amazon
B Trench Elementary Differential Equations Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.