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

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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 ' ) ( ; )), (...
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L10_ODE - MA2213 Lecture 10 ODE Topics Importance p 367-368...

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