2 CHAPTER 2 First-Order Differential Equations 2.1 Solution Curves Without the Solution2.2 Separable Variables2.3 Linear Equations2.4 Exact Equations2.5 Solutions by Substitutions2.6 A Numerical Solution2.7 Linear Models2.8 Nonlinear Models2.9 Systems: Linear and Nonlinear Models
3 2.1 Solution Curves Without the Solution xySlope=Short tangent segments suggest the shape of the curve輪廓
4 2.1 Solution Curves Without the Solution The general first-order differential equation has the formF(x, y, y’)=0or in the explicit formy’=f(x,y)Note that, a graph of a solution of a first-order differential equation is called a solution curve or an integral curve of the equation.On the other hand, the slope of the integral curve through a given point (x0,y0) is y’(x0).
5 2.1 Solution Curves Without the Solution A drawing of the plane, with short line segments of slope drawn at selected points , is called a direction field or a slope field of the differential equation .The name derives from the fact that at each point the line segment gives the direction of the integral curve through that point. The line segments are called lineal elements.
6 2.1 Solution Curves Without the Solution Plotting Direction Fields1st Stepy’=f(x,y)=C=constant curves of equal inclination2nd StepAlong each curve f(x,y)=C, draw lineal elements direction field3rd StepSketch approximate solution curves having the directions of the lineal elements as their tangent directions.
7 2.1 Solution Curves Without the Solution If the derivative dy/dx is positive (negative) for all x in an interval I, then the differentiable function y(x) is increasing (decreasing) on I.
8 2.1 Solution Curves Without the Solution DEFINITION: autonomous DEA differential equation in which the independentvariable does not explicitly appear is knownas an autonomous differential equation.For example, a first order autonomous DE hasthe formDEFINITION: critical pointA critical point of an autonomous DE is a realnumber c such that f(c) = 0.Another name for critical point is stationary point orequilibrium point.If c is a critical point of an autonomous DE, then y(x) = c is aconstant solution of the DE.
9 2.1 Solution Curves Without the Solution DEFINITION: phase portraitA one dimensional phase portrait of anautonomous DE is a diagram which indicatesthe values of the dependent variable for which y isincreasing, decreasing or constant.Sometimes the vertical line of the phase portrait iscalled a phase line.
10 DEFINITION: Separable DE 2.2 Separable VariablesDEFINITION: Separable DEA first-order differential equation of the formis said to be separable or tohave separable variables.(Zill, Definition 2.1, page 44).
11 2.2 Separable VariablesMethod of SolutionIf represents a solution
12 2.2 Separable Variables domain , , The Natural Logarithm The natural exponential functiondomain
13 DEFINITION: Linear Equation 2.3 Linear EquationsDEFINITION: Linear EquationA first-order differential equation of the formis said to be a linear equation.(Zill, Definition 2.2, page 51).When homogeneousOtherwise it is non-homogeneous.
15 2.2 Separable EquationsA differential equation is called separable if it can be written asSuch that we can separate the variables and writeWe attempt to integrate this equation
16 2.2 Separable Equations Example 1. is separable. Write as Integrate this equation to obtainor in the explicit formWhat about y=0 ? Singular solution !
17 2.2 Separable Equations Example 2. is separable, too. We write Integrate the separated equation to obtainThe general solution isAgain, check if y=-1 is a solution or not ?it is a solution, but not a singular one, since it is a special case of the general solution
18 2.3 Linear Differential Equations A first-order differential equation is linear if it has the formMultiply the differential equation by to getNow integrate to obtainThe function is called an integrating factor for the differential equation.
19 2.3 Linear Differential Equations Linear: A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables.Non-linear: Differential equations that do not satisfy the definition of linear are non-linear.Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.
20 2.3 Linear Differential Equations Exampleis a linear DE. P(x)=1 and q(x)=sin(x), both continuous for all x.An integrating factor isMultiply the DE by to getOrIntegrate to getThe general solution is
21 2.3 Linear Differential Equations ExampleSolve the initial value problemIt can be written in linear formAn integrating factor is forMultiply the DE by to getOrIntegrate to get ,thenforFor the initial condition, we needC=17/4 the solution of the initial value problemis