TATM59 Ordinary Differential Equations 2007

Study Guide@lith

Valid for year : 2007

TATM59	Ordinary Differential Equations, 6 ECTS credits. /Ordinära differentialekvationer/ For: COM M
	Prel. scheduled hours: 60 Rec. self-study hours: 100
	Area of Education: Science Subject area: Mathematics
	Advancement level (G1, G2, A): G2
	Aim: The course will give knowledge about properties of ordinary differential equations which are essential in applied mathematics and mechanics. Important applications are in the study of vibrations and other dynamic problems in the theory of elasticity, including analysis of stability. After completing this course the students should be able to: Understand and use theorems of existence and uniqueness for ordinary differential equations and systems of such equations. Give examples of initial value problems with multiple solutions. Solve explicitely certain simple differential equations of the first order, such as linear, homogeneous and separable equations and so called Bernoulli equations. Use principles of solutions for equations of higher order, in particular linear equations. Analyze the Wronski determinant and its properties. Use the Wronski determinant for analysing the linear indepency of homogeneous solutions and for finding solution formulas for linear inhomogeneous equations. Use the method of variation of parameters and reduction of the order for solving nonhomogeneous linear problems. Use a matrix formalism for solving simple systems of differential equations, in particular equations with constant coefficients. Explain the concept of fundamental matrix solution and use it for finding solution formulas of nonhomogeneous linear systems. Explain and give examples of the concept of autonomous systems. Use the method of linearization in order to investigate stability. Give an account of the concept Liapunov function and use it for stability investigations. Analyse boundary value problems for ordinary differential equations and describe how they appear when solving partial differentiqal equations. Solve inhomogeneous boundary value problems by the aid of Green functions. Give an account of eigen value problems for Sturm-Liouville equations and the concept convergence in the mean square sense. Expand solutions after orthogonal Sturm-Liouville systems in simple cases.
	Prerequisites: (valid for students admitted to programmes within which the course is offered) Linear algrebra, Calculus in one and several variables Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
	Organisation: Teaching is done in seminars.
	Course contents: Existence and uniqueness for linear and non-linear differential equations. Construction of solutions by Euler's method, some numerical methods of solving ordinary differential equations. Linear systems with constant coefficients. Exponentials of matrices. Homogeneous and inhomogeneous systems. Some aspects of Sturm-Liouville theory and special functions. Autonomous systems. Phase portrait. Linearisation of autonomous systems. Liapounov's method for stability analysis. Periodic solutions.
	Course literature: W.E. Boyce, R.C.DiPrima: Elementary Differential Equations and Boundary Problems. Wiley & Sons.
	Examination:
	Written examination	4 p	/	6 ECTS


Course language is Swedish. Department offering the course: MAI. Director of Studies: Göran Forsling Examiner: Lars-Erik Andersson Link to the course homepage at the department Course Syllabus in Swedish

Linköping Institute of Technology
Contact: TFK , val@tfk.liu.se Last updated: 09/03/2007