L10_ODE - MA2213 Lecture 10 ODE Topics Importance p....

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA2213 Lecture 10 ODE Topics Importance p. 367-368 Introduction to the theory p. 368-373 Analytic solutions p. 368-372 Existence of solutions p. 372 Direction fields p. 376-379 umerical methods Forward Euler p. 383 Richardson’s extrapolation formula p. 391 Systems of equations p. 432 Two point boundary value problems p. 442 Importance “Differential equations are among the most important mathematical tools used in producing odels of physical and biological sciences, and ngineering.” They can be classified into: rdinary : )) ( , ( f ) ( ' x Y x x Y = have 1 independent variable ) ( ) ( ), ( ) ( t x t y t y t x = − = & & artial : have > 1 independent variable 2 2 2 2 2 2 y u x u t u ∂ ∂ + ∂ ∂ = ∂ ∂ 2 2 2 2 y u x u t u ∂ ∂ + ∂ ∂ = ∂ ∂ heat equation wave equation Analytic Solutions Integration ∫ + = ⇒ = c x d x g x Y x g x Y ) ( ) ( ) ( ) ( ' Integrating Factors eparation of Variables c x x x x x Y x x x + + − = ⇒ = cos 2 sin ) 2 ( ) ( cos ) ( 2 2 ) ( ) ( ) ( ) ( ' x b x Y x a x Y + = ( ) ) ( ) ( ), ( ) ( ) ( ) ( x a dx x d x b e x Y e dx d x x = = − − λ λ λ ∫ ∫ + = + = − δ λ λ λ dx x a x c dx x b e e z x x ) ( ) ( ], ) ( [ ) ( ) ( ) ( ( ) K x Y x rY x Y / ) ( 1 ) ( ) ( ' − = ⇒ = − dx r K x Y x x dY ) / ) ( 1 )( ( ) ( ) 1 )( ( ) ( ) ( − + = x r x r e Y K e Y K x Y Existence of Solutions heorem 8.1.3 (page 372) Let and be continuous functions of ) , ( f z x z z x ∂ / ) , ( f x ) ( x Y and at all points ) , ( z x in some neighborhood of ). , Y x Then there is a unique function efined on some interval ], , [ α α + − x x satisfying ' ) ( ; )), (...
View Full Document

This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.

Page1 / 20

L10_ODE - MA2213 Lecture 10 ODE Topics Importance p....

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online