1、Section 7.12Examples of Differential EquationsSuppose that a plane curve passed through the point(1,2)inthe xOy plane.The slope of the tangent at any point(x,y)to the curve is2x.Find the equation of the curve.Solution By the geometric meaning of derivatives,the desiredcurve y=f(x)should satisfy12,|2
2、.xdyxydx.Integrating on both sides of the 1st equation with respect to x,we obtaindifferentialequationInitialcondition3Examples of Differential EquationsSolution(continued)Substituting the initial condition to the last21C 1CTherefore,the equation of the desired curve is21.yxxy21yxO12equation,we have
3、Finish.22yxdxxC where C is an arbitrary constant.It can be evaluated by the initial condition.Suppose that a plane curve passed through the point(1,2)inthe xOy plane.The slope of the tangent at any point(x,y)to the curve is2x.Find the equation of the curve.Example 2 Suppose that a particle with mass
4、 m falls freely froma position of height H,with initial velocity V0.If we neglect the resistance of air,find the relationship between the height H and time t while the particle is falling.4Examples of Differential EquationsHh(t)SolutionDenote the initial time when the particle startsto fall by t=0,a
5、nd denote the height of the particle at any time t in the process of falling by h=h(t).By Newtons second law,h should satisfy the followingequation2222i.e.d hd hmmggdtdt 000|,|tthH hV differentialequationInitialconditionsExample 2 Suppose that a particle with mass m falls freely froma position of he
6、ight H,with initial velocity V0.If we neglect the resistance of air,find the relationship between the height H and time t while the particle is falling.5Examples of Differential EquationsHh(t)Solution(continued)22d hgdt Integrating both sides of the last equation twice,we have21212hgtC tC Finish.201
7、()-.2h tgtV tHSubstituting the initial condition to the aboveequation yields6Basic ConceptsDefinition(Differential equation)An equation is called a differentialequation(微分方程微分方程)if it contains the derivative or differential of an unknown function.2dyxdx and22d hgdt are both differential equations.De
8、finition A differential equation in which the unknown function y is a univariate function,is called an ordinary differential equation(常微分方程常微分方程)and will be referred as differential equation(微分方微分方程程).Example:The order of the highest order derivative of the unknown function in the equation is called
9、 the order(阶阶)of the equation.7Basic Concepts2,dyxdx 0,ydxxdy220dyyxydxFirst-Order22,d hgdt 33,xyyye3()yyxSecond-OrderExample:The general form of a first order differential equation may be expressed by(,)yF x yand F(x,y)is a function which depends on the independent variable x and the dependent vari
10、able y.8Basic ConceptsDefinition (Solution,General Solution,Initial Conditions and Particular Solution)If the solution contains arbitrary constants and the number of the independent constants just equals the order of the equation,then this solution is called the general solution(通解通解)of the equation
11、.If all the arbitrary constants in a solution have been determined,then the solution is called a particular solution(特解特解)of the equation.If a function y=f(x)satisfies a given differential equation,then the function y=f(x)is called a solution(解解)of the equation.The additional conditions are called t
12、he initial conditions(初始条初始条件件)of the equation.9Fundamental Concepts of Differential EquationsNote that the general solution may not be the total solutions.2()0.y yx(1)0.y yExample 10Geometric Interpretation of the First Order Differential Equation2dyxdx 2yxC0|1xy 21yx11Geometric Interpretation of t
13、he First Order Differential EquationSlope Fields:Viewing Solution CurvesEach time we specify an initial condition y(x0)=y0 for the solution of a differential equation ,the solution curve(graph of the solution)is required to pass through the point(x0,y0)and to have slope f(x0,y0)there.(,)yf x y12Geom
14、etric Interpretation of the First Order Differential EquationSlope Fields:Viewing Solution CurvesWe can picture the slopes graphically by drawing short line segments of slope f(x,y)at selected points(x,y)in the region of the xy-plane that constitutes the domain of f.Each segment has the same slope as the solution curve through(x,y)and so is tangent to the curve there.We see how the curves behave by following these tangents.13Slope Fields
