MATH 634 Theor Ord Diff Eqs BYU

MATHEMATICS 6342 FALL SEMESTER 1999 Classroom: Class Time: Instructor: 2308 SFLC. 12:0012:50 MWF. Chris Grant, 283 TMCB, 378-4105, grant@math.byu.edu. A webpage for this class is located at: http:/www.math.byu.edu/~grant/courses/m634/f99/index.html

Lecture Notes on Ordinary Dierential Equations Christopher P. Grant 1 ODEs and Dynamical Systems Lecture 1 Math 634 8/30/99 Ordinary Dierential Equations An ordinary dierential equation (or ODE) is an equation involving derivatives of an unknown qu

ODEs and Dynamical Systems Lecture 1 Math 634 8/30/99 Ordinary Dierential Equations An ordinary dierential equation (or ODE) is an equation involving derivatives of an unknown quantity with respect to a single variable. More precisely, suppose j, k

Existence of Solutions Lecture 2 Math 634 9/1/99 Approximate Solutions Consider the IVP x = f (t, x) x(t0 ) = a, (1) where f : dom(f ) R R n R n is continuous, and (t0 , a) dom(f ) is constant. The Fundamental Theorem of Calculus implies that (

Uniqueness of Solutions Lecture 3 Math 634 9/3/99 Uniqueness If more than continuity of f is assumed, it may be possible to prove that x = f (t, x) x(t0 ) = a, (1) has a unique solution. In particular Lipschitz continuity of f (t, ) is sucient. (Re

Picard-Lindelf Theorem o Lecture 4 Math 634 9/8/99 Theorem The space C([a, b]) of continuous functions from [a, b] to R n equipped with the norm f is a Banach space. Denition Two dierent norms 1 and 2 on a vector space X are equivalent if there exi

Intervals of Existence Lecture 5 Math 634 9/10/99 Maximal Interval of Existence We begin our discussion with some denitions and an important theorem of real analysis. Denition Given f : D R R n R n , we say that f (t, x) is locally Lipschitz conti

Dependence on Parameters Lecture 6 Math 634 9/13/99 Parameters vs. Initial Conditions Consider the IVP x = f (t, x) x(t0 ) = a, and the paramterized IVP x = f (t, x, ) x(t0 ) = a, (2) (1) where R k . We are interested in studying how the solutio

Constant Coecient Linear Equations Lecture 7 Math 634 9/15/99 Linear Equations Denition Given f : R Rn Rn , we say that the rst-order ODE x = f (t, x) (1) is linear if every linear combination of solutions of (1) is a solution of (1). Denition Gi