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1、The integrating factor method (Sect. 2.1)IOverview of differential equations.ILinear Ordinary Differential Equations.IThe integrating factor method.IConstant coefficients.IThe Initial Value Problem.IVariable coefficients.Read:IThe direction field. Example 2 in Section 1.1 in the Textbook.ISee direct
2、ion field plotters in Internet. For example, see:http:/math.rice.edu/edfield/dfpp.html This link is given in our class webpage.Overview of differential equations.DefinitionA differential equation is an equation, where the unknown is a function, and both the function and its derivative appear in the
3、equation.Remark:There are two main types of differential equations:IOrdinary Differential Equations (ODE): Derivatives with respect to only one variable appear in the equation.Example: Newtons second law of motion: ma = F.IPartial differential Equations (PDE): Partial derivatives of two or more vari
4、ables appear in the equation.Example: The wave equation for sound propagation in air.Overview of differential equations.Example Newtons second law of motion is an ODE: The unknown is x(t), the particle position as function of time t and the equation isd2 dt2x(t) =1 mF(t,x(t),with m the particle mass
5、 and F the force acting on the particle.Example The wave equation is a PDE: The unknown is u(t,x), a function that depends on two variables, and the equation is2 t2u(t,x) = v22 x2u(t,x),with v the wave speed. Sound propagation in air is described by a wave equation, where u represents the air pressu
6、re.Overview of differential equations.Remark:Differential equations are a central part in a physical description of nature:IClassical Mechanics:INewtons second law of motion. (ODE)ILagranges equations. (ODE)IElectromagnetism:IMaxwells equations. (PDE)IQuantum Mechanics:ISchr odingers equation. (PDE)
7、IGeneral Relativity:IEinstein equation. (PDE)IQuantum Electrodynamics:IThe equations of QED. (PDE).The integrating factor method (Sect. 2.1).IOverview of differential equations.ILinear Ordinary Differential Equations.IThe integrating factor method.IConstant coefficients.IThe Initial Value Problem.IV
8、ariable coefficients.Linear Ordinary Differential EquationsRemark:Given a function y : R R, we use the notationy0(t) =dy dt(t).Definition Given a function f : R2 R, a first order ODE in the unknown function y : R R is the equationy0(t) = f (t,y(t).The first order ODE above is called linear iffthere
9、exist functions a,b : R R such that f (t,y) = a(t)y + b(t). That is, f islinear on its argument y, hence a first order linear ODE is given byy0(t) = a(t)y(t) + b(t).Linear Ordinary Differential EquationsExampleA first order linear ODE is given byy0(t) = 2y(t) + 3.In this case function a(t) = 2 and b
10、(t) = 3. Since these function do not depend on t, the equation above is called of constantcoefficients.ExampleA first order linear ODE is given byy0(t) = 2 ty(t) + 4t.In this case function a(t) = 2/t and b(t) = 4t. Since these functions depend on t, the equation above is called of variablecoefficien
11、ts.The integrating factor method (Sect. 2.1).IOverview of differential equations.ILinear Ordinary Differential Equations.IThe integrating factor method.IConstant coefficients.IThe Initial Value Problem.IVariable coefficients.The integrating factor method.Remark:Solutions to first order linear ODE ca
12、n be obtained using the integrating factor method.Theorem (Constant coefficients)Given constants a,b R with a 6= 0, the linear differential equation y0(t) = ay(t) + bhas infinitely many solutions, one for each value of c R, given byy(t) = c eat+b a.Remark:A proof is given in the Lecture Notes. Here
13、we present the main idea of the proof, showing and exponential integrating factor. In the Lecture Notes it is shown that this is essentially the only integrating factor.The integrating factor method.Main ideas of the Proof:Write down the differential equation asy0(t) + ay(t) = b.Key idea: The left-h
14、and side above is a total derivative if we multiply it by the exponential eat. Indeed,eaty0(t) + aeaty(t) = beat eaty0(t) +?eat?0y(t) = beat.This is the key idea, because the derivative of a product implies ?eaty(t)?0= beat.The exponential eatis called an integrating factor. Indeed, we can now integ
15、rate the equation!eaty =Z beatdt + c eaty =b aeat+ c y(t) = c eat+b a.The integrating factor method.Example Find all functions y solution of the ODE y0= 2y + 3.Solution: The ODE is y0= ay + b with a = 2 and b = 3.The functions y(t) = ceat+b a, with c R, are solutions.We conclude that the ODE hasinfi
16、nitely many solutions, given byy(t) = c e2t3 2,c R.Since we did one integration, it is reasonable that the solution contains a constant of integration, c R.3/2c 00ytc = 0Verification: c e2t= y + (3/2), so 2c e2t= y0, therefore weconclude that y satisfies the ODE y0= 2y + 3.CThe integrating factor method (Sect. 2.1).IOverview of differential equations.ILinear Ordinary Differential Equations.IThe integrating factor method.IConstant coefficients.IT
