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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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