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# Class Note for MATH 250A with Professor Lega at UA

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 29 views.

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Date Created: 02/06/15

Calculus and Differential Equations l MATH 250 A Introduction to differential equations A simple differential equation 7 o Is there a function which is equal to its derivative 0 Yes 9 No o Is such a function unique 0 Yes 9 No d o The equation d y y is an example of an ordinary differential X equation 0 The independent variable is X and the dependent variable is y o The above equation is a first order autonomous and linear differential equation Introducticui to d Calculus and al Equations itial equations Introducticui to differential equations Calculus and D rential Equations Solutions of a differential equation a An explicit solution of an ordinary differential equation dy dx is a function yx such that when substituted into the differential equation both sides are found to be identical WW 0 In general a given differential equation will have a family of solutions involving one or more parameters 0 Applying initial or boundary conditions often leads to the selection of one of these solutions a We will now turn to two important questions what are differential equations used for and how do we study them a To address the first questions we now look at examples of differential equations or systems thereof Calculus and Differential Equations lntroducticui to differential equations The nonlinear pendulum The equation of motion for the nonlinear pendulum is given by d20 d0 ml 7 7mg Sn07CE7 where o 0 and t are variables 0 m g and c are parameters 9 Most of these quantities are defined on the figure except c which measures friction Sketch of a pointmass pendulum Introducticui to d rential aluations Calculus and Dif ntial Equations The RLC c39 The series RLC circuit consists of a resistor of resistance R an inductor of inductance L a capacitor of R capacitance C and a power source of voltage Vt VC The charge q across the capacitor I gt satisfies the differential equation C dzq dq 1 L V0 dt2 Rdt Eq Image by Omeganon released under a Creative Commons Almbulion ShareAlike license version 30 2 5 2 u and 1 u and the current in the circuit is given dq b t y dt The clas 39 SIR model The classic SIR model reads d5 E O SN d a as e i dR E 7 57 where 5 and R represent the numbers of susceptible infectious and recovered or removed indIVIduals in a population of size N The parameter 1 measures the average number of Penned goats in a Village Wllhln a region investigated for a Rm Valley fever outbreak in Saudi Arabia Picture 8362 Public positive contacts per susceptible per unit of time and 3 measures the rate at which indIVIduals recover Heath Image Library The the dynamics of a Viral infection such as hepatitis B or C may be described by the folloWIng model M A Nowak et al Proc Natl Acad Sci USA 93 439874402 1996 K Aeaxebvx dt biXan dt d we v dt N The variable X represents the number Transmission electron micrograph showmg of uninfected cells Y is the number of hepatiu Vll39lOnS of an unknown Strain Plame 8153 Public Healm Image infected cells and V is the Viral load or Library number of free Virions in the body How do we study 6 Sometimes we can solve a differential equation In this class MATH 250 A 84 B we Will learn how to solve first and second order linear equations and systems of first order linear equations as well as some first order nonlinear equations 0 If initial conditions are known one can solve a differential equation or a system of differential equations numerically We Will learn a simple numerical method to solve a differential equation and also use more advanced algorithms in MATLAB 9 Before trying to solve a differential equation or launching into a numerical exploration of its properties one needs to know whether solutions egtltist and if so whether they are unique We Will see theorems that guarantee egtltistence and uniqueness of solutions to differential equations differential equations continued 0 In many situations especially when one deals with nonlinear differential equations one cannot find explicit solutions 9 In this case one can nevertheless understand the dynamics of a differential equations by looking at special solutions and at their stability 9 The qualitative theory of dynamical systems discussed in MATH 454 provides a way of understanding the behavior of a system of differential equations as well as the bifurcations that occur when one or more parameters are changed We will briefly address some of these issues 0 Partial differential equations discussed in MATH 322 MATH 422 and MATH 456 are differential equations describing the dynamics of systems with two or more independent variables Introducticm to differential equations Calculus and D rential Equations What we will do next 0 We will start with the simplest type of differential equations dy X dx 0 Chapter 1 of Differential Equations book Reading assignment Sections 11 12 and 13 o Solving such differential equations involves integration so we will introduce various methods of integration discussed in the Calculus book a We will then consider autonomous differential equations of the dy 7 form a 7 gy 0 Chapter 2 of the Differential Equations book 0 Ideas of stability and bifurcations 0 One more technique of integration partial fractions In troducticui to d mai equations Calculus and al Equations i I at we will do neXt continued e We will then turn to general first order differential equations dy E XVy 0 Chapter 3 of the Differential Equations book a Graphical analysis symmetries scalings and numerical solutions 0 Existence and uniqueness of solutions 0 Finally we will look at various methods of solution for first order differential equations 0 Chapters 4 and 5 of the Differential Equations book 0 Separation of variables and equations with homogeneous coefficients 9 First order linear differential equations and Bernouilli39s equation Introducticm to differential equations Calculus and Differential Equations

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