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Unformatted text preview: Math 219, Lecture 4 Ali Devin Sezer October 7, 2010 Contents 1 First order one dimensional differential equations 1 1.1 Linear equations and integrating factors . . . . . . . . . . . . 2 1.2 Exact and separable equations . . . . . . . . . . . . . . . . . 3 1.2.1 Homogeneous equations . . . . . . . . . . . . . . . . . 6 2 Differences between linear and nonlinear equations 6 2.1 Bernoulli Equations . . . . . . . . . . . . . . . . . . . . . . . 7 3 Autonomous equations and population dynamics 7 1 First order one dimensional differential equa tions These are equations of the form y ′ ( t ) = f ( t,y ( t )) (1) where f : R 2 → R . The goal is to find a differentiable function y defined on some interval [ a,b ] satisfying (1). What we know about (1) so far: 1. If we are given an initial condition y ( t ) = y (2) in addition to (1) and if we know that f and ∂f ∂y are continuous in a rectangle t R 1 ≤ t ≤ t + R 1 , y R 2 ≤ y ≤ y + R 2 the 1 existence and uniqueness theorem guarantees that the initial value problem consisting of (1) and (2) has a unique solution. 2. The Euler approximation method y ( t ) = y , y ( t + nδ ) = y ( t +( n 1) δ )+ δf ( t + δ ( n 1) ,y ( t +( n 1) δ )) can be used to find approximate solutions to the initial value problem; smaller δ , better the approximation. 3. Finally, we can draw the differential equation as a direction field, which gives an overall idea as to how the solutions behave. In today’s lecture we will learn three new methods that allow one to solve (1) exactly when f has a certain form. These five methods are: 1. if f is linear one can use integrating factors , 2. (1) is called “exact” if it is obtained by differentiating an implicit rela tion. There is a simple way to check if the equation is exact, in which case it is possible to write back the original implicit relation. Some times even if (1) is not exact, it can be multiplied with an integrating factor to make it exact....
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 Spring '10
 uguz
 Differential Equations, Linear Equations, Equations, Factors, Derivative, Elementary algebra, 0 g

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