Course syllabus - Differential Equations, foundation course

Objectives

The course aims at introducing the fundamental qualitative and quantitative methods in use to analyze and solve ordinary differential equations and applications thereof.

Learning outcomes

At the end of a passed course, the student is expected to be able to

- analyse and solve ordinary differential equations (ODE) of first order including concepts and methods such as existence, uniqueness, phase portrait, separability, linearity, exactness and substitution.

- analyse and solve linear ODE including concepts and methods such as complimentary solution, particular solution, reduction of order, and power series solution about an ordinary point

- analyse and solve nonlinear but reducible ODE of order 2

- apply the Laplace transform for solving initial value problems (IVP) of linear differential- and integral equations

- solve plane systems of first order ODE with constant coefficients, and be able to analyse their stabilities and to sketch their phase portraits

- analyse nonlinear ODE of order 2 and plane autonomous systems of first order nonlinear ODE, all with respect to stability in neighbourhoods of stationary points, and in certain cases be able to linearize the systems considered

- by autonomous systems of ODE describe dynamical macroscopic courses of events assuming that occurring number quantities are defined on intervals and that underlying microscopic processes are instantaneous

Course content

- Differential equations (DE) in general: order, existence of solution, particular solution, general solution, unique solution, initial value problems (IVP)

- ODE of first order: phase portrait, orthogonal trajectories, separability, linearity, exactness, substitution techniques (for homogeneous DE, for Bernoulli DE, for DE with powers of ax+by+c), introducing examples of applications

- Linear ODE: BVP, existence of an unique solution, linear independent solutions, Wronskian, homogenous solution, reduction of order, particular solution, variation of parameters, general solution, power series solution about an ordinary point

- Some special ODE of order ≥ 2: Euler equations, nonlinear but in order reducible ODE

- The Laplace transform: existence, standard transforms, inverse transforms, transforms of derivatives, translations, Heaviside’s step function, derivatives of transforms, transforms of integrals, especially convolutions, Dirac’s delta distribution, solutions of differential- and integral equations

- Plane system of first order linear ODE: BVP, existence of a unique solution, linear independent solutions, Wronskian, fundamental matrix, homogeneous solution, particular solution, general solution, phase portrait

- Plane autonomous systems of first order nonlinear ODE and nonlinear ODE of order 2: stability for linear systems, local stability for nonlinear systems, analysis at stationary points, linearization, long-range behaviour

Tuition

Teaching is given in the form of lectures and classes.

Specific requirements

Single Variable Calculus 7.5 credits and Vector Algebra 7.5 credits or the equivalent.

Examination

Assigned problems (INL1), 2,5 credits, marks Fail (U), Pass (G)
Written and/or oral examination (TEN1), 5 credits, marks Fail (U), 3, 4, 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