Course syllabus - Differential Equations and Transform Methods

Objectives

The course aims at introducing the fundamental qualitative and quantitative methods used in analysis of differential equations and difference equations, and associated transforms and applications.

Learning outcomes

At the end of the course the student is expected to be able to
- analyse and solve ordinary differential equations of first order including concepts/methods such as uniqueness, phase portraits, separability, linearity, exactness, and substitution techniques.
- analyse and solve linear differential equations including concepts/methods such as fundamental solutions, particular solutions, reduction, and substitution techniques.
- apply the Laplace transform for finding solutions to linear differential- and integral equations with given initial conditions, and to be able to apply the Z-transform for solving linear difference equations with constant coefficients.
- solve linear systems of first order differential equations with constant coefficients, and to be able to analyse their stabilities and to obtain their phase portraits.
- analyse the most common types of nonlinear plane autonomous systems of first order differential equations with respect to stability in neighbourhoods of stationary points, and in possible cases to be able to linearize such systems.
- with common approximating assumptions, describe dynamical behaviours within (closed) ecosystems.
- explain what orthogonal, orthonormal, and complete systems of functions are, and to be able to apply the system of trigonometric functions on functions on bounded intervals.
- with separation techniques and those knowledges which are specified in the learning objective 7, solve boundary- and intial-value problems including the one-dimensional heat equation, the ditto wave equation, and the Laplace equation in two dimensions.

Course content

First order differential equations. Linear differential equations of second order. The Laplace transform. Difference equations. The Z-transform. The Fourier transform. Dirac's delta pulse. Systems of differential equations. Qualitative methods for non-linear differential equations. Analysis at critical points. Long range behavior. Stability. Existence and uniqueness theorems. Fourier series, orthogonal bases. Partial differential equations: Separation of variables. Applications on ordinary and partial differential equations.

Tuition

Teaching is by lectures, assigned problems and classes.

Specific requirements

At least 60 credits in the technical, natural sciences, business administration or economics areas where Calculus II 7,5 credits and Linear Algebra 7,5 credits or equivalent are included.

Examination

Assigned problems (INL1), 2.5 credits, marks Pass (G)
Written and/or oral examination (TEN1), 5 credits, marks 3, 4 or 5

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Pass with distinction, Pass with credit, Pass, Fail