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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
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
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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