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L10_ODE - MA2213 Lecture 10 ODE Topics Importance p 367-368...

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MA2213 Lecture 10 ODE
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Topics Importance p. 367-368 Introduction to the theory p. 368-373 Numerical methods Forward Euler p. 383 Richardson’s extrapolation formula p. 391 Analytic solutions p. 368-372 Existence of solutions p. 372 Direction fields p. 376-379 Systems of equations p. 432 Two point boundary value problems p. 442
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Importance “Differential equations are among the most important mathematical tools used in producing models of physical and biological sciences, and engineering.” They can be classified into: Ordinary : )) ( , ( f ) ( ' x Y x x Y = have 1 independent variable ) ( ) ( ), ( ) ( t x t y t y t x = - = Partial : have > 1 independent variable 2 2 2 2 2 2 y u x u t u + = wave equation 2 2 2 2 y u x u t u + = heat equation
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Analytic Solutions Integration + = = c x d x g x Y x g x Y ) ( ) ( ) ( ) ( ' Integrating Factors Separation of Variables c x x x x x Y x x x g + + - = = cos 2 sin ) 2 ( ) ( cos ) ( 2 2 ) ( ) ( ) ( ) ( ' x b x Y x a x Y + = ( 29 ) ( ) ( ), ( ) ( ) ( ) ( 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 Y x x ) ( ) ( ], ) ( [ ) ( ) ( ) ( ( 29 K x Y x rY x Y / ) ( 1 ) ( ) ( ' - = = - dx r K x Y x Y x dY ) / ) ( 1 )( ( ) ( ) 1 )( 0 ( ) 0 ( ) ( - + = x r x r e Y K e Y K x Y
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Existence of Solutions Theorem 8.1.3 (page 372) Let and be continuous functions of 0 0 0 0 ' ) ( ; )), ( , ( f ) ( Y x Y x x x x Y x x Y = + - = α α ) , ( f z x z z x / ) , ( f x 1 ) 0 ( ; ] ) ( [ 2 ) ( 2 ' = = Y x Y x x Y ) ( x Y and z at all points ) , ( z x in some neighborhood of ). , ( 0 0 Y x Then there is a unique function defined on some interval ], , [ 0 0 α α + - x x satisfying Example 8.1.4 (p. 372) The initial value problem admits the solution 1 1 , 1 1 ) ( 2 < < - - = x x x Y
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Direction Fields
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