ODE_Handout_2x2-1 - Ordinary differential equations Linear differential equations and systems Nonhomogeneous linear equations and systems Chapters

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Unformatted text preview: Ordinary differential equations Linear differential equations and systems Nonhomogeneous linear equations and systems Chapters 1-2-4: Ordinary Differential Equations Sections 1.1, 1.7, 2.2, 2.6, 2.7, 4.2 & 4.3 Chapters 1-2-4: Ordinary Differential Equations Ordinary differential equations Linear differential equations and systems Nonhomogeneous linear equations and systems Definitions Existence and uniqueness of solutions 1. Ordinary differential equations An ordinary differential equation of order n is an equation of the form d n y dx n = f x , y , dy dx , . . . , d n- 1 y dx n- 1 . (1) A solution to this differential equation is an n-times differentiable function y ( x ) which satisfies (1). Example: Consider the differential equation y- 2 y + y = 0 . What is the order of this equation? Are y 1 ( x ) = e x and y 2 ( x ) = x e x solutions of this differential equation? Are y 1 ( x ) and y 2 ( x ) linearly independent? Chapters 1-2-4: Ordinary Differential Equations Ordinary differential equations Linear differential equations and systems Nonhomogeneous linear equations and systems Definitions Existence and uniqueness of solutions Initial and boundary conditions An initial condition is the prescription of the values of y and of its ( n- 1)st derivatives at a point x , y ( x ) = y , dy dx ( x ) = y 1 , . . . d n- 1 y dx n- 1 ( x ) = y n- 1 , (2) where y , y 1 , ... y n- 1 are given numbers. Boundary conditions prescribe the values of linear combinations of y and its derivatives for two different values of x . In MATH 254 , you saw various methods to solve ordinary differential equations. Recall that initial or boundary conditions should be imposed after the general solution of a differential equation has been found. Chapters 1-2-4: Ordinary Differential Equations Ordinary differential equations Linear differential equations and systems Nonhomogeneous linear equations and systems Definitions Existence and uniqueness of solutions 2. Existence and uniqueness of solutions Equation (1) may be written as a first-order system dY dx = F ( x , Y ) (3) by setting Y = y , dy dx , d 2 y dx , · · · , d n- 1 y dx n- 1 T . Existence and uniqueness of solutions: if F in (3) is continuously differentiable in the rectangle R = { ( x , Y ) , | x- x | < a , || Y- Y || < b , a , b > } , then the initial value problem dY dx = F ( x , Y ) , Y ( x ) = Y , has a solution in a neighborhood of ( x , Y ). Moreover, this solution is unique....
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This note was uploaded on 12/01/2009 for the course MATH 322 taught by Professor Staff during the Spring '08 term at University of Arizona- Tucson.

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ODE_Handout_2x2-1 - Ordinary differential equations Linear differential equations and systems Nonhomogeneous linear equations and systems Chapters

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