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ODE_Handout_2x2-1

# ODE_Handout_2x2-1 - Ordinary dierential equations Linear...

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Ordinary di ff erential equations Linear di ff erential equations and systems Nonhomogeneous linear equations and systems Chapters 1-2-4: Ordinary Di ff erential Equations Sections 1.1, 1.7, 2.2, 2.6, 2.7, 4.2 & 4.3 Chapters 1-2-4: Ordinary Di ff erential Equations Ordinary di ff erential equations Linear di ff erential equations and systems Nonhomogeneous linear equations and systems Definitions Existence and uniqueness of solutions 1. Ordinary di ff erential equations An ordinary di ff erential 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 di ff erential equation is an n -times di ff erentiable function y ( x ) which satisfies (1). Example: Consider the di ff erential 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 di ff erential equation? Are y 1 ( x ) and y 2 ( x ) linearly independent? Chapters 1-2-4: Ordinary Di ff erential Equations Ordinary di ff erential equations Linear di ff erential 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 0 , y ( x 0 ) = y 0 , dy dx ( x 0 ) = y 1 , . . . d n - 1 y dx n - 1 ( x 0 ) = y n - 1 , (2) where y 0 , y 1 , ... y n - 1 are given numbers. Boundary conditions prescribe the values of linear combinations of y and its derivatives for two di ff erent values of x . In MATH 254 , you saw various methods to solve ordinary di ff erential equations. Recall that initial or boundary conditions should be imposed after the general solution of a di ff erential equation has been found. Chapters 1-2-4: Ordinary Di ff erential Equations Ordinary di ff erential equations Linear di ff erential 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 di ff erentiable in the rectangle R = { ( x , Y ) , | x - x 0 | < a , || Y - Y 0 || < b , a , b > 0 } , then the initial value problem dY dx = F ( x , Y ) , Y ( x 0 ) = Y 0 , has a solution in a neighborhood of ( x 0 , Y 0 ). Moreover, this solution is unique.

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