linear algebra, and the central ideas of direct methods for the numerical solution of dense linear systems as described in standard texts such as [7], [105],or[184]. Solving nonlinear ordinary differential equations using the NDM. Nonlinear Differential Equations and The Beauty of Chaos 2 Examples of nonlinear equations 2 ( ) kx t dt d x t m =− Simple harmonic oscillator (linear ODE) More complicated motion (nonlinear ODE) ( )(1 ()) 2 ( ) kx t x t dt d x t m =− −α Other examples: weather patters, the turbulent motion of fluids Most natural phenomena are essentially nonlinear. Our approach is to focus on a small number of methods and treat them in depth.

and not a number. Linear vs Nonlinear Di erential Equations An ODE for y= y(t) is linear if it can be written in the form a n(t)y(n) + a n 1(t)y (n 1) + + a 3(t)y(3) + a 2(t)y00+ a 1(t)y0+ a 0(t)y= g(t) where the ‘coe cient’ functions g(t);a 0(t);:::;a n(t) can be any functions of t, (including the zero function), and where we are using the following notation for It will, in a few pages, provide a link between nonlinear and linear … Linearheremeansthattheunknownfunction,oraderivativeofit,ismultiplied byanumberoraknownfunction. 3

Allotherdiﬀerentialequationsarenon-linear. Article (PDF Available) ... is an excellent mathematical tool for solving linear and nonlinear differential equation.

Chapter & Page: 43–4 Nonlinear Autonomous Systems of Differential Equations You may have encountered this creature (or its determinant) in other courses involving “two functions of two variables” or “multidimensional change of variables”. Though this book is written in a ﬁnite-dimensional setting, we In a linear diﬀerential equation, all terms involving the unknown functions are linear in the unknown functions or their derivatives.